Module 1 Energy Methods n Structural Analyss Verson 2 CE IIT, Kharagpur
Lesson 6 Engesser s Theorem and Truss Deflectons by Vrtual Work Prncples Verson 2 CE IIT, Kharagpur
Instructonal Objectes After readng ths lesson, the reader wll be able to: 1. State and proe Crott-Engesser theorem. 2. Dere smple expressons for calculatng deflectons n trusses subjected to mechancal loadng usng unt-load method. 3. Dere equatons for calculatng deflectons n trusses subjected to temperature loads. 4. Compute deflectons n trusses usng unt-load method due to fabrcaton errors. 6.1 Introducton In the preous lesson, we dscussed the prncple of rtual work and prncple of rtual dsplacement. Also, we dered unt load method from the prncple of rtual work and unt dsplacement method from the prncple of rtual dsplacement. In ths lesson, the unt load method s employed to calculate dsplacements of trusses due to external loadng. Intally the Engesser s theorem, whch s more general than the Castglano s theorem, s dscussed. In the end, few examples are soled to demonstrate the power of rtual work. 6.2 Crott-Engesser Theorem The Crott-Engesser theorem states that the frst partal derate of the * U expressed n terms of appled forces s complementary stran energy ( equal to the correspondng dsplacement. F j U * n = ajk Fk = uj (6.1 F j k = 1 For the case of ndetermnate structures ths may be stated as, U F j * = 0 (6.2 Note that Engesser s theorem s ald for both lnear and non-lnear structures. When the complementary stran energy s equal to the stran energy (.e. n case of lnear structures the equaton (6.1 s nothng but the statement of Castglano s frst theorem n terms of complementary stran energy. Verson 2 CE IIT, Kharagpur
In the aboe fgure the stran energy (area OACO s not equal to complementary stran energy (area OABO Area OACO = U u = F du (6.3 0 Dfferentatng stran energy wth respect to dsplacement, du = F (6.4 du Ths s the statement of Castglano s second theorem. Now the complementary energy s equal to the area enclosed by OABO. U * F = 0 u df (6.5 Dfferentatng complementary stran energy wth respect to force F, * du = u df (6.6 Verson 2 CE IIT, Kharagpur
Ths ges deflecton n the drecton of load. When the load dsplacement relatonshp s lnear, the aboe equaton concdes wth the Castglano s frst theorem gen n equaton (3.8. 6.3 Unt Load Method as appled to Trusses 6.3.1 External Loadng In case of a plane or a space truss, the only nternal forces present are axal as the external loads are appled at jonts. Hence, equaton (5.7 may be wrtten as, n L δp Pds δ F u = (6.7 EA j j j= 1 0 wheren, δ Fj s the external rtual load, u j are the actual deflectons of the truss, L P δ P s the rtual stress resultant n the frame due to the rtual load and ds 0 EA s the actual nternal deformaton of the frame due to real forces. In the aboe equaton L, E, A respectely represent length of the member, cross-sectonal area of a member and modulus of elastcty of a member. In the unt load method, δ F j = 1 and all other components of rtual forces δ F ( = 1,2,..., j 1, j + 1,..., n are zero. Also, f the cross sectonal area A of truss remans constant throughout, then ntegraton may be replaced by summaton and hence equaton (6.7 may be wrtten as, u j = m = 1 (δp j P L (6.8 where m s the number of members, (δ P j s the nternal rtual axal force n P member due to unt rtual load at j and ( L s the total deformaton of EA member due to real loads. If we represent total deformaton by, then m u j = (δ P j Δ (6.9 = 1 Δ where, Δ s the true change n length of member due to real loads. 6.3.2 Temperature Loadng Due to change n the enronmental temperature, the truss members ether expand or shrnk. Ths n turn produces jont deflectons n the truss. Ths may be Verson 2 CE IIT, Kharagpur
calculated by equaton (6.9. In ths case, the change n length of member calculated from the relaton, Δ s Δ = αtl (6.10 where α s the co-effcent of thermal expanson member, member and T s the temperature change. L s the length of 6.3.3 Fabrcaton Errors and Camber Sometmes, there wll be errors n fabrcatng truss members. In some cases, the truss members are fabrcated slghtly longer or shorter n order to prode camber to the truss. Usually camber s proded n brdge truss so that ts bottom chord s cured upward by an equal to ts downward deflecton of the chord when subjected to dead. In such nstances, also, the truss jont deflecton s calculated by equaton (6.9. Here, Δ = (6.11 e where, e s the fabrcaton error n the length of the member. e s taken as poste when the member lengths are fabrcated slghtly more than the actual length otherwse t s taken as negate. 6.4 Procedure for calculatng truss deflecton 1. Frst, calculate the real forces n the member of the truss ether by method of jonts or by method of sectons due to the externally appled forces. From ths P L determne the actual deformaton ( Δ n each member from the equaton. E A Assume tensle forces as poste and compresse forces as negate. 2. Now, consder the rtual load system such that only a unt load s consdered at the jont ether n the horzontal or n the ertcal drecton, where the deflecton s sought. Calculate rtual forces ( δ P j n each member due to the appled unt load at the j-th jont. 3. Now, usng equaton (6.9, ealuate the j-th jont deflecton u j. 4. If deflecton of a jont needs to be calculated due to temperature change, then determne the actual deformaton ( Δ n each member from the equaton Δ = αtl. The applcaton of equaton (6.8 s shown wth the help of few problems. Verson 2 CE IIT, Kharagpur
Example 6.1 Fnd horzontal and ertcal deflecton of jont C of truss ABCD loaded as shown n Fg. 6.2a. Assume that, all members hae the same axal rgdty. The gen truss s statcally determnate one. The reactons are as shown n Fg 6.2b along wth member forces whch are determned by equatons of statc equlbrum. To ealuate horzontal deflecton at C, apply a unt load as shown n Fg 6.2c and ealuate the rtual forces δ P n each member. The magntudes of nternal forces are also shown n the respecte fgures. The tensle forces are shown as +e and compresse forces are shown as e. At each end of the bar, arrows hae been drawn ndcatng the drecton n whch the force n the member acts on the jont. Verson 2 CE IIT, Kharagpur
Horzontal deflecton at jont C s calculated wth the help of unt load method. Ths may be stated as, 1 u H c ( δp c P L = (1 For calculatng horzontal deflecton at C,, apply a unt load at the jont C as shown n Fg.6.2c. The whole calculatons are shown n table 6.1. The calculatons are self explanatory. u c Verson 2 CE IIT, Kharagpur
Table 6.1 Computatonal detals for horzontal deflecton at C Member Length L / A E P (δ P (δp E A unts m m/kn kn kn kn.m AB 4 4/AE 0 0 0 BC 4 4/AE 0 0 0 CD 4 4/AE -15-1 60/AE DA 4 4/AE 0 0 0 AC 4 2 4 2 /AE 5 2 2 40 2 /AE 60 + 40 AE 60 + 40 2 116.569 ( 1( u H C = = (Towards rght (2 AE Vertcal deflecton at jont C 1 u c ( δp c P L = In ths case, a unt ertcal load s appled at jont C of the truss as shown n Fg. 6.2d. Table 6.2 Computatonal detals for ertcal deflecton at C Member Length L / A E P (δ P (δp unts m m/kn kn kn kn.m AB 4 4/AE 0 0 0 BC 4 4/AE 0 0 0 CD 4 4/AE -15-1 60/AE DA 4 4/AE 0 0 0 AC 4 2 4 2 /AE 5 2 0 0 60 AE 60 60 ( 1( u C = = (Downwards (4 AE P L P L 2 (3 Verson 2 CE IIT, Kharagpur
Example 6.2 Compute the ertcal deflecton of jont b and horzontal dsplacement of jont D of the truss shown n Fg. 6.3a due to a Appled loadng as shown n fgure. 0 b Increase n temperature of 25 C n the top chord BD. Assume 1 5 2 α = per C, E = 2.00 10 N / mm. The cross sectonal areas of the 75000 members n square centmeters are shown n parentheses. Verson 2 CE IIT, Kharagpur
Verson 2 CE IIT, Kharagpur
The complete calculatons are shown n the followng table. Table 6.3 Computatonal detals for example 6.2 Mem L L / A E P P H (δ (δp Δ t = αtl (δp P L H (δ P PL H (δ (δp t Δ t P Δ unts m (10-5 m/kn kn kn kn m (10-3 kn.m (10-3 kn.m (10-3 kn.m (10-3 kn.m ab 5 1.0-112.5-0.937 +0.416 0 1.05-0.47 0 0 ab 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0 bc 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0 Bc 5 1.0 +37.5-0.312-0.416 0-0.12-0.16 0 0 BD 6 2.0-67.5-0.562 +0.500 0.002 0.76-0.68-1.13 1 cd 5 1.0 +37.5 +0.312 +0.416 0 0.12 0.16 0 0 cd 3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0 de 3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0 De 5 1.0-112.5-0.312-0.416 0 0.35 0.47 0 0 Bb 4 2.0 +60.0 1 0 0 1.2 0 0 0 Dd 4 2.0 +60.0 0 0 0 0 0 0 0 4.38 0.68-1.13 1 Verson 2 CE IIT, Kharagpur
a Vertcal deflecton of jont b Applyng prncple of rtual work as appled to an deal pn jonted truss, ( δ P PL 1 m j δ Fu j j = j= 1 = 1 EA (1 For calculatng ertcal deflecton at b, apply a unt rtual load δ F b = 1. Then the aboe equaton may be wrtten as, 1 u 1 Due to external loads b ( δp P L = (2 u b = + 0.00438 KNm 1 KN = 0.00438 m = 4.38 mm 2 Due to change n temperature t ( 1( u = ( δ P Δ b 0.001125 KN. m u t b = = 0. 00113m 1 KN u t b = 1. 13 mm t b Horzontal dsplacement of jont D 1 Due to externally appled loads u H D 1 u H b H ( δp P L = + 0.00068 KNm = = 0.00068 m 1 KN = 0.68 mm Verson 2 CE IIT, Kharagpur
2 Due to change n temperature Ht H ( 1( u = ( δ P Δ D 0.001 KN. m D = = 0. 001m 1 KN u Ht u Ht D = 1. 00 mm t Summary In ths chapter the Crott-Engessor s theorem whch s more general than the Castglano s theorem has been ntroduced. The unt load method s appled statcally determnate structure for calculatng deflectons when the truss s subjected to arous types of loadngs such as: mechancal loadng, temperature loadng and fabrcaton errors. Verson 2 CE IIT, Kharagpur