Calculate by hand the deformation of joint 2 of the following simple beam using the Virtual Work Method.
Hint for solution
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| body | --uriencoded--$$ \normalsize Δ =\int \frac %7BMM%5e'%7D%7BEI%7D $$ |
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According to the method; the integral should be considered separately for the bars between 1-2 and 2-3 nodes.
Detailed solution
E= Modulus of elasticity A= Cross-sectional area I= Cross-section moment of inertia
M= Moment due to external loading M'= Moment due to unit loading Δ= Deformation
Step 1 Finding Moment function by Unit Loading
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Image AddedQ: unit load D 1 , D 3 : Support responses M'(2): 2 DN torque value L: Element length
M' 1-2 (x): 1-2 Torque function M 2-3 (x): 2- 3 Torque function ΣM1,dn: Total torque relative to 1 DN.
ΣD: Total balance
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =0 $$ |
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =Q*(L/2)-D_3*L=0 $$ |
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =1*(4/2)-D_3*4=0 $$ |
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| body | $$ \normalsize D_3=0.5 tf $$ |
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| body | $$ \normalsize ΣD =0 $$ |
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| body | $$ \normalsize ΣD = D_1 + D_3 - Q = 0 $$ |
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| body | $$ \normalsize ΣD = D_1 + 0.5 - 1 = 0 $$ |
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| body | $$ \normalsize D_1=0.5 tf $$ |
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| body | --uriencoded--$$ \normalsize M'_%7B1-2%7D(0) = 0 $$ |
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| body | --uriencoded--$$ \normalsize M'_%7B2-3%7D(0) = 0 $$ |
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| body | --uriencoded--$$ \normalsize M'_%7B1-2%7D(2) = M'_%7B2-3%7D(2) = D_1*(L/2) = D_2*(L/2) $$ |
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| body | --uriencoded--$$ \normalsize M'_%7B1-2%7D(2) = M'_%7B2-3%7D(2) =0.5*(4/2) = 0.5*(4/2) = 1 $$ |
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| body | --uriencoded--$$ \normalsize M'_%7B1-2%7D(x=L/2) = M'_%7B2-3%7D(x=L/2) = M'(2) =1 tf.m $$ |
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| body | $$ \normalsize M'(2) =1 tf.m $$ |
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Moment Function 1-2 :
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| body | --uriencoded--$$ \normalsize M'_%7B1-2%7D(x) = 0.5x $$ |
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Moment Function 2-3 :
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| body | --uriencoded--$$ \normalsize M'_%7B2-3%7D(x) = -0.5*x+2 $$ |
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Step 2: Finding the Moment Function Based on the Given Load
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Image Addedq: Distributed load D 1 , D 3 : Support responses M(2): 2 DN torque value L: Element length
M(x): Torque function ΣM 1,dn : Total torque with respect to 1 DN.
ΣD: Total balance
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =0 $$ |
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =(q*L)*(L/2)-D_3*L=0 $$ |
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| body | --uriencoded--$$ \normalsize ΣM_%7B1,dn%7D =(2*4)*(4/2)-D_3*4=0 $$ |
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| body | $$ \normalsize D_3=4 tf $$ |
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| body | $$ \normalsize ΣD =0 $$ |
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| body | $$ \normalsize ΣD = D_1 + D_3 - q*L = 0 $$ |
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| body | $$ \normalsize ΣD = D_1 + 4 - 2*8 = 0 $$ |
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| body | $$ \normalsize D_1=4 tf $$ |
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| body | $$ \normalsize M(0) =0 tfm $$ |
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| body | $$ \normalsize M(4) =0 tfm $$ |
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| body | $$ \normalsize M(2) =D_1*(L/2) - q*(L/2)*(L/4) = D_2*(L/2)- q*(L/2)*(L/4)tfm $$ |
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| body | $$ \normalsize M(2) =4*(2)-2*(4/2)*(4/4) = 4*(2)-2*(4/2)*(4/4) =4tfm $$ |
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| body | $$ \normalsize M(2) =4 tfm $$ |
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Moment Function :
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| body | --uriencoded--$$ \normalsize M(x) = -x%5e2 + 4x $$ |
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Step 3: Multiplying the M and M' Moment Functions in the Previous Steps and Getting the Integral
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| body | --uriencoded--$$ \normalsize Δ =\int \frac %7BM(x)M%5e'(x)%7D%7BEI%7Ddx $$ |
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| body | --uriencoded--$$ \normalsize Δ = Δ_%7B1-2%7D + Δ_%7B2-3%7D $$ |
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| body | --uriencoded--$$ \normalsize Δ =\int \frac %7BM_%7B1-2%7DM_%7B1-2%7D%5e'%7D%7BEI%7D + \int \frac %7BM_%7B2-3%7DM_%7B2-3%7D%5e'%7D%7BEI%7D $$ |
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Integral for 1-2 Bar
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| body | --uriencoded--$$ \normalsize Δ_%7B1-2%7DEI = \int_0%5e%7BL/2%7D %7B(-x%5e2+4x)0.5x dx%7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B1-2%7DEI = \int_0%5e%7B2%7D %7B(-0.5x%5e3+2x%5e2) dx%7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B1-2%7DEI = \frac %7B-0.5x%5e4%7D 4 \right%7C_0%5e2 + \frac %7B2x%5e3%7D 3 \right%7C_0%5e2 %7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B1-2%7DEI = -2 + 5.333 = 3.333 %7D $$ |
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Integral for 2-3 Bar
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| body | --uriencoded--$$ \normalsize Δ_%7B2-3%7DEI = \int_%7BL/2%7D%5e%7BL%7D %7B(-x%5e2+4x)(-0.5x+2) dx%7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B2-3%7DEI = \int_2%5e%7B4%7D %7B(0.5x%5e3-4x%5e2+8x) dx%7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B2-3%7DEI = \frac %7B0.5x%5e4%7D 4 \right%7C_2%5e4 - \frac %7B4x%5e3%7D 3 \right%7C_2%5e4 + \frac %7B8x%5e2%7D 2 \right%7C_2%5e4 %7D $$ |
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| body | --uriencoded--$$ \normalsize Δ_%7B2-3%7DEI = 30 -74.667 +48 = 3.333 %7D $$ |
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| body | --uriencoded--$$ \normalsize ΔEI = Δ_%7B1-2%7DEI + Δ_%7B2-3%7DEI = 3.333 + 3.333 %7D $$ |
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| body | --uriencoded--$$ \normalsize ΔEI = 6.666 %7D $$ |
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| body | --uriencoded--$$ \normalsize E = 360t/cm%5e2 %7D $$ |
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| body | --uriencoded--$$ \normalsize I = (1/12)*(12*12%5e3) = 1728cm%5e4 %7D $$ |
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| body | --uriencoded--$$ \normalsize EI = E * I = 360 * 1728 = 622080tf.cm%5e2 %7D $$ |
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| body | --uriencoded--$$ \normalsize EI = 622080tf.cm%5e2 = 62.208 tf.m%5e2 %7D $$ |
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| body | --uriencoded--$$ \normalsize ΔEI = 6.666 %7D $$ |
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| body | --uriencoded--$$ \normalsize EI = 62.208 tf.m%5e2 %7D $$ |
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| body | --uriencoded--$$ \normalsize Δ = 6.666/EI = 6.666/62.208 = 0.107 m %7D $$ |
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Deformation of node 2 = 0.107m