Chapter7 - Chapter 7 Matrix Formulation of the Consistent...

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© K. H. Ha - ENGR 343 Chapter 7 - v2.0 7.1 Chapter 7 Matrix Formulation of the Consistent Deformation Method 1. Beams . ................................................................................................. 2 1.1 The Model . ..................................................................................... 2 1.2 Nodal Equilibrium Equation. .............................................................. 3 1.3 Selection of Redundants. .................................................................. 4 1.4 Member Forces Q = B X X + Q L ........................................................... 4 1.5 Compatibility Equation . .................................................................... 5 1.6 Solution. ........................................................................................ 6 1.7 Computation. .................................................................................. 6 Exercise 1 . ............................................................................................. 8 2. Trusses. ................................................................................................ 8 2.1 The Model . ..................................................................................... 9 2.2 Nodal Equilibrium Equation. .............................................................. 9 2.3 Selection of Redundants. ................................................................ 10 2.4 Member Forces Q = B X X + Q L ......................................................... 10 2.5 Compatibility Equation . .................................................................. 11 2.6 Solution. ...................................................................................... 12 2.7 Computation. ................................................................................ 12 Exercise 2 . ........................................................................................... 13 3. Frames . .............................................................................................. 14 3.1 The Model . ................................................................................... 14 3.2 Nodal Equilibrium Equation. ............................................................ 15 3.3 Selection of Redundants. ................................................................ 16 3.4 Member Forces Q = B X X + Q L ......................................................... 16 3.5 Compatibility Equation . .................................................................. 17 3.6 Solution. ...................................................................................... 18 3.7 Computation. ................................................................................ 18 Exercise 3 . ........................................................................................... 20
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© K. H. Ha - ENGR 343 Chapter 7 - v2.0 7.2 1. Beams Matrix formulation for solution is generally applicable to all systems including beams, trusses and frames. The example beam in Fig.7.1 will be used for illustration of the steps. 40 kN 10 kN/m 20 kN m 10 kN.m × I = 236 10 6 mm 4 a b c d e L =4m 2m 4m k = EI/L 3 × E = 200 10 6 kPa Figure 7.1 1.1 The Model The model in Fig.7.2 includes: ± Bending action in the members ab , bd and de . ± Axial action in the spring at b . ± Two nodal directions, N = 2. The nodal forces R for the given loading are zero. a b c d e Q 1 Q 2 Q 3 Q 4 R 1 R 2 Figure 7.2 The members’ flexibility relation is, for the sequence of Q in Fig.7.2:
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© K. H. Ha - ENGR 343 Chapter 7 - v2.0 7.3 + + = 0 6 10 24 10 24 20 16 40 0 1 6 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2 6 3 2 4 3 2 1 2 4 3 2 1 L L L L EI Q Q Q Q L EI L q q q q or q = f Q + q o where q o are due to member loading (Table 2.1). 1.2 Nodal Equilibrium Equation The nodal equilibrium equation has the following form (see Eq.1.3): 1 1 1 × × × × + = N M M N N W Q b R (7.1) where Q is the vector of independent member forces to account for the dominant actions in the members, and W is the vector of nodal forces contributed by the member loading. The equilibrium matrix b is rectangular with M > N for statically indeterminate systems. The degree of statical indeterminacy is: I = M – N = 4 - 2 = 2 Q 1 R 1 V 1 Q 2 V 2 40 kN 20 kN m 10 kN/m 10 kN.m R 2 Q 3 Q 4 a c d e b Figure 7.3 For the example beam (see Fig. 7.3): 3 2 2 4 2 1 4 2 1 4 2 1 1 20 2 40 20 2 40 Q Q R L Q L Q L Q Q L L Q L Q Q V V R + = + + + = + + + = + + =
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© K. H. Ha - ENGR 343 Chapter 7 - v2.0 7.4 ( 7 . 2 ) + ⎡− = 0 / 20 20 0 1 1 0 1 0 / 1 / 1 4 3 2 1 2 1 L Q Q Q Q L L R R (7.3) Thus, = ⎡− = 0 20 20 0 1 1 0 1 0 / 1 / 1 L L L W b (7.4) 1.3 Selection of Redundants Since I = 2, we need to introduce 2 redundants. Among many choices, we may choose: 4 2 2 1 Q X Q X = = ( 7 . 5 ) 1.4 Member Forces Q = B X X + Q L Combining Eqs.7.3 & 5: +
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Chapter7 - Chapter 7 Matrix Formulation of the Consistent...

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