Underpinning and retention by S. Thorburn, G.S. Littlejohn
By S. Thorburn, G.S. Littlejohn
A variety of underpinning tools exist, and selecting a selected technique or choice of tools is the task of the engineer. in step with the underpinning strategy is the necessity to maintain the affected constitution plus the encircling flooring and/or constructions. This booklet will supply recommendation on easy methods to selected the right kind process.
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DMN T g ¼ ½g10 ; . ; gln ; . ; g2N T h ¼ ½h10 ; . ; hlm ; . ; h2M T E ¼ ½E00 ; . ; Emn ; . ; EMN T i ¼ ½i10 ; . ; iln ; . ; i2N T j ¼ ½j10 ; . ; jlm ; . ; j2M T ð2:61Þ 54 2 Modiﬁed Fourier Series and Rayleigh-Ritz Method Superscript T represents the transposition operator. Substituting Eq. 59) into Eq. 48) results in À Á Lxy Cxy þ Lx Cx þ Ly Cy À x2 Mxy Cxy þ Mx Cx þ My Cy ¼ 0 ð2:62Þ where 2 L11 Hi 6 L21 Hi 6 Li ¼6 6 L31 Hi 4 L41 Hi L51 Hi 2 M11 Hi 60 6 Mi ¼ 6 60 4 M41 Hi 0 L12 Hi L22 Hi L32 Hi L42 Hi L52 Hi L13 Hi L23 Hi L33 Hi L43 Hi L53 Hi L14 Hi L24 Hi L34 Hi L44 Hi L54 Hi 0 M22 Hi 0 0 M52 Hi 0 0 M33 Hi 0 0 M14 Hi 0 0 M44 Hi 0 3 A 6B 7 6 7 7 Cxy ¼6 6 C 7; 4D5 E 2 3 L15 Hi L25 Hi 7 7 L35 Hi 7 7; L45 Hi 5 L55 Hi ði ¼ xy; x; yÞ 3 0 M25 Hi 7 7 7; 0 7 5 0 M55 Hi 2 3 a 6c 7 6 7 7 Cx ¼6 6 e 7; 4g5 i ði ¼ xy; x; yÞ 2 3 b 6d7 6 7 7 Cy ¼6 6f 7 4h5 j ð2:63Þ ð2:64Þ ð2:65Þ In the same way, substituting Eq.
1 Combining Eqs. 13) results in 1 w000 ðLÞ À w000 ð0Þ X 2D11 m 000 000 þ D11 ½ðÀ1Þ w ðLÞ À w ð0Þ cos km x L L m¼1 1 X 2D11 ½ðÀ1Þm w0 ðLÞ À w0 ð0Þk2m þ L m¼1 À Á 4 2 þ D11 km À x I0 Am cos km x ¼ x2 I0 ð2:15Þ Obviously, it is a big challenge to obtain the natural frequencies and determine the expansion coefﬁcients from Eq. 15). Alternatively, one may prefer to expand the beam displacement w(x) in the form of Fourier sine series. In such case wðxÞ ¼ 1 X Am sin km x; ð2:16Þ 0\x\L m¼1 Then w0 ðxÞ ¼ wðLÞ À wð0Þ L 1 X 2 ½ðÀ1Þm wðLÞ À wð0Þ þ Am km cos km x; þ L m¼1 w00 ðxÞ ¼ À 1 X 2 m¼1 w000 ðxÞ ¼ L ½ðÀ1Þm wðLÞ À wð0Þkm þ Am k2m sin km x; ð2:17Þ 0 x L 0\x\L ð2:18Þ w00 ðLÞ À w00 ð0Þ L !
Any point P in the middle surface of the sectorial plate is deﬁned by the vector r ¼ r cos h Á i þ r sin h Á j ð1:73Þ Thus, the Lamé parameters of sectorial plates can be obtained as @r @r ¼ ¼ cos h Á i þ sin h Á j @a @r @r A ¼ ¼ jcos h Á i þ sin h Á jj ¼ 1 @r ð1:74Þ @r @r ¼ ¼ Àr sin h Á i þ r cos h Á j @b @h @r B ¼ ¼ jÀr sin h Á i þ r cos h Á jj ¼ r @h ð1:75Þ (c) Cylindrical shells Let α = x and β = θ, where α is parallel to the axis of revolution and β is in the circumferential direction as shown in Fig.