Calculation of Stiffness in Structural elements

A1 we are going to use the stiffness influence coefficint to solve the problem.

the 3x3 stiffness matrix of the structure is known as equilibrium condition.

the DOF is 3

Step 1

U1=1, U2=0, U3=0

stiffness matrix calculate as

K11=24EIc/h^3

K12=6EIc/h^2

K13=6EIc/h^2

Step 2

U1=0, U2=1, U3=0

K21=6EIc/h^2

K22=4EIc/h+4EIb/L

K23=2EIb/L

Step 3

U1=0, U2=0, U3=1

stiffness matrix calculated by

K31=6EIc/h^2

K32=2EIb/L

K33=4EIc/h+4EIb/L

therefore

K matrix of [K11 K12 K13

K21 K22 K23

K31 K32 K33]

K=EIc/h^3 matrix of [24 6h 6h {u1 ={fs

6h 6h^2 h^2 u2 0

6h 6h^2 6h^2 u3} 0}

Equation

24u1+6hu2+6hu3=fs 1

6hu1+6h^2u2+h^2u3=0 2

6hu1+h^2u2+6h^2u3=0 3

Equation can be solved by elimination method or static condensation method

[6h^2 h^2 {u2 = 6h{U1

h^2 6h^2] u3 U1}

[u2= {6h^2 h^2 6h{U1 = 6/7h[1 u1

u3] h^2 6h^2] (invese) 6h 1]

fs=24EIc/h^3-EIc/h^3*6/7h (6h 6h)1by1)u1

96/7EIc/h^3u1

A2

a3 consider the propped cantilever. the beams are made of the same steel section and have lengths as shown in the diagram

the lateral stiffness we need to add both stiffness of beam

for simply supported beam latral stiffness = 48Ei/L^3

for cantilever supported beam lateral stiffness = 3EI/L^3

add both beam stiffness

K=(48EI/L^3+3EI/L^3)

K=51EI/L^3

A4 effective stiffness of cantilever beam is K=48EI/L^3

effective stiffness of simply supported beam is K=EI/L^3

effectve stiffness of simply supported beam is K=EI/L^3+3EI/l^3

=6EI/L^3