Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey arge defection theory introduces changes into the strain-dispacement reation and vertica equiibrium, but eaves the constitutive equation and horizonta equiibrium unchanged. The kinematica reation, Eq. (2.68) acquires now a new term due to finite rotations of beam eement. ɛ = du + 1 2 The definition of curvature has aso a noninear rotation term ( ) dw 2 new term (6.1) d 2 w κ = [ 2 ( ) ] dw 2 3/2 (6.2) 1 + The square of the sope can be arge, as compared with the term du and must be retained in Eq. (6.1). At the same time the square of the sope (beam rotation) are sma compared to unity. Why? This is expained in Fig. (6.1), where the square of the sope is potted against the sope. Figure 6.1: The significance of the square of the sope term. At θ = 1 rad = 57 degrees the two terms in the denominator of Eq. (6.2) are equa. However, the theory of moderatey arge defections are vaid up to θ = 1.175 rad. The term θ 2 amounts to.3, which is negigibe compared to unity. Therefore the curvature is defined in the same way as in the theory of sma defections κ = d2 w 2 (6.3) 6-1
Structura Mechanics 2.8 Lecture 6 Semester Yr It was shown in Lecture 3 that the equation of equiibrium in the horizonta direction is not affected by the finite rotation. Therefore, we infer from Eq. (3.77) that the axia force is either constant or zero N = constant (6.4) The vertica equiibrium, given by Eq. (3.79) has a new noninear term d 2 M 2 new term + N d2 w 2 +q = (6.5) Finay, the easticity aw is unaffected by finite rotation N = EAɛ (6.6a) M = EIκ (6.6b) The soution to the couped probem depends on the boundary conditions in the horizonta direction. Referring to Fig. 5.1, two cases must be considered: Case 1, beam free to side, N =, u. Case 2, beam fixed, u =, N. 6.2 Soution for a Beam on Roer Support Consider first case 1. From the constitutive equation, zero axia force beams that there is no extension of the beam axis, ɛ =. Then, from Eq. (6.1) du = 1 2 ( ) dw 2 (6.7) At the same time, the noninear term in the vertica equiibrium vanishes and the beam response is governed by the inear differentia equation EI d4 w = q(x) (6.8) 4 which is identica to the one derived for the infinitesima defections. As an exampe, consider the pin-pin supported beam under mid-span point oad. From Eqs. (5.54) and (5.55), the defection profie is [ w(x) = w o 3 x ( x ) ] 3 4 (6.9) and the sope is dw = w [ ( o x ) ] 2 3 12 (6.1) 6-2
Structura Mechanics 2.8 Lecture 6 Semester Yr where w o is the centra defection of the beam. Now, Eq. (6.7) can be used to cacuate reative horizonta dispacement u. Integrating Eq. (6.7) in the imits (, ) gives The resut of the integration is du = u 1 = u() u() = u = 2 ( ) dw 2 (6.11) u 7 w2 o (6.12) In order to get a physica sense of the above resut, the vertica and horizonta dispacements are normaized by the thickness h of the beam u h = 7 /h ( wo ) 2 (6.13) h For a beam with h = 21, the resut is poted in Fig. (6.2). u h = 1 3 ( wo ) 2 (6.14) h Figure 6.2: Siding of a beam from the roer support. It is seen that the amount of siding in the horizonta direction can be very arge compared to the thickness. To summarize the resuts, the roer supported beam can be treated as a cassica beam even though the dispacements and rotations are arge (moderate). The soution of the inear differentia equation can then be used a posteriori to determine the magnitude of siding. The anaysis of fuy restrained beam is much more interesting and difficut. This is the subject of the next section. 6-3
Structura Mechanics 2.8 Lecture 6 Semester Yr 6.3 Soution for a Beam with Fixed Axia Dispacements The probem is soved under the assumption of simpy-supported end condition, and the ine oad is distributed accordingy to the cosine function.the beam is restrained in the axia direction. There is a considerabe strengthening effect of the beam response due to finite rotations of beam eements. The axia force N (non-zero this time) is cacuated from Eq. (6.6) with Eq. (6.1) N = EA [ du + 1 2 ( ) ] dw 2 (6.15) From Eq. (6.4) we know that N is constant but unknown. In order to make use of the kinematic boundary conditions, et us integrate both sides of Eq. (6.15) with respect to x N 1 = u() u() + EA 2 ( ) dw 2 (6.16) Using the boundary conditions for u, the axia force is reated to the square of the sope by N EA = 1 2 ( ) dw 2 (6.17) In order to determine the defected shape of the beam, consider the equiibrium in the vertica direction given by Eq. (6.5) Dividing both sides by ( EI) one gets EI d4 w 4 + N d2 w 2 + q = (6.18) where d 4 w 4 λ2 d2 w 2 = q o πx cos EI λ 2 = N EI (6.19) (6.2) The roots of the characteristic equations are,, ±λ. Therefore the genera soution of the homogeneous equation is w g = C o + C 1 x + C 2 cosh λx + C 3 sinh λx (6.21) As the particuar soution of the inhomogeneous equation we can try w p (x) = C cos πx d 2 w p 2 = πx π2 C cos 2 d 4 w p 4 = π4 πx C cos 4 (6.22a) (6.22b) (6.22c) 6-4
Structura Mechanics 2.8 Lecture 6 Semester Yr Substituting the above soution to the governing equation (6.18) one gets [ π 4 π2 C λ2 4 2 C P ] o cos πx = (6.23) EI The above soution satisfy the differentia equation if the ampitude C is reated to input parameters and the unknown tension N C = π 2 2 q o EI ) = (λ 2 + π2 2 q o ( π ) 4 ( π ) 2 (6.24) EI + N The genera soution of Eq. (6.18) is a sum of the particuar soution of the inhomogeneous equation w p and genera soution of the homogeneous equation, w g w(x) = w g + w p (6.25) There are five unknowns, C o, C 1, C 2, C 3 and N and five equations. Four boundary conditions for the transverse defections w =, d 2 w 2 = at x = ± 2 (6.26) and equation (6.17) reating the horizonta and vertica response. The determination of the integration constants is straightforward. Note that the probem is symmetric. Therefore the soution shoud be an even function 1 of x. The terms C 1 x and C 3 sinh λx are odd functions. Therefore the respective coefficients shoud vanish C 1 = C 3 = (6.27) w(x) = C o + C 2 cosh λx + C cos πx (6.28) The remaining two coefficients are determined ony from the boundary conditions at one side of the beam C o + C 2 cosh λ 2 = w(x = 2 ) = d 2 w 2 = x= 2 The soution of the above system is C 2 λ 2 cosh λ 2 = The sope of the defection curve, cacuated from Eq. (6.28) is (6.29) C o = C 2 = (6.3) dw = C π sin πx 1 The function is even when f( A) = f(a). The function is odd when f( A) = f(a). (6.31) 6-5
Structura Mechanics 2.8 Lecture 6 Semester Yr Expressing N in terms of x in Eq. (6.17) gives ( ) I λ 2 = 1 A 2 Combining the above two equations one gets ( ) I λ 2 = 1 A 2 ( ) dw 2 (6.32) ( Cπ sin πx ) 2 (6.33) or after integration λ 2 I A = 1 ( π ) 2 4 C2 (6.34) 2 Recaing the definition of λ, the membrane force N becomes a quadratic function of the defection ampitude N = EA ( π ) 2 4 C2 (6.35) The membrane force can be eiminated between Eqs. (6.24) and (6.35) to give the cubic equation for the defection ampitude C C + C 3 A 4I = q ( ) o 4 (6.36) EI π To get a better sense of the contribution of various terms, consider a beam of the square cross-section h h, for which I = h 12, A = h2, A I = 12 h 2 (6.37) Aso, the ventra defection is dimensionaized with respect to the beam thickness w o = C h w o + 3w 3 o = ( qo ) ( ) 4 (6.38) Eh πh The present soution is exact and invoves a information about the materia (E), oad intensity (q o ), ength () and cross-sectiona dimension. The distribution of ine oad and boundary conditions are refected in the specific numerica coefficients in the respective terms. In order to get a physica insight about the contributions of a terms in the above soution, consider two imiting cases: (i) Pure bending soution for which N dw =. (ii) Pure membrane (string, cabe) soution with zero fexura resistance (bending rigidity, EI ). 6-6
Structura Mechanics 2.8 Lecture 6 Semester Yr (i) The bending soution is obtained by dropping the cubic term in Eqs. (6.36) or (6.38) C = q o 4 EI 1 π 4 (6.39) where the coefficient π 4 = 97.4. this resut for the sinusoida distribution of the ine oad shoud be compared with the uniform ine oad (coefficient 77) and point oad (coefficient 48). (ii) The membrane soution is recovered by negecting the first inear term C 3 = q o 4 EI 4 π 4 (6.4) The pot of the fu bending/membrane soution and two imiting cases is shown in Fig. (6.3). Figure 6.3: Comparison of a bending and membrane soution for a beam. The question is at which magnitude of the centra defection reative to the beam thickness the non-dimensiona oad q o is the same. This is the intersection of the straight ine Eh with the third order paraboa. By eiminating the oad parameter between Eqs. (6.39) and (6.4) one gets C 2 = 4I A = 4ρ2 A A = 4ρ2 (6.41) where ρ is the radius of gyration of the cross-section. For a square cross-section I h C = 2ρ = 2 ρ = 2 4 12h 2 = h =.58h (6.42) 3 It is concuded that the transition from bending to membrane action occurs quite eary in the beam response. As a rue of thumb, the bending soution in the beam restrained from axia motion is restricted to defections equa to haf of the beam thickness. If defections are arger, the membrane response dominate. For exampe, if beam defection reaches three 6-7
Structura Mechanics 2.8 Lecture 6 Semester Yr thicknesses, the contribution of bending and membrane action is 3:81. In the upper imit of the appicabiity of the theory of moderatey arge defection of beams C = 1h, the contribution of bending resistance is ony.3% of the membrane strength. The rapid transition from bending to membrane action is ony present for axiay restrained beams. If the beam is free to side in the axia direction, no membrane resistance is deveoped and oad is aways ineary reated to defections. The above eegant cosed-form soution was obtained for the particuar sinusoida distribution of ine oad, which coincide with the defected shape of the beam. For an arbitrary oading, ony approximate soutions coud be derived. One of such soution method, appicabe to the broad cass of non inear probems for pates and shes is caed the Gaerkin method. 6.4 Gaerkin Method of Soving Non-inear Differentia Equation Beris Gaerkin, a Russian scientist, mathematician and engineer was active in the first forty ears of the 2 th century. He is an exampe of a university professor who appied methods of structura mechanics to sove engineering probems. At that time (Word War I), the unsoved probem was moderatey arge defections of pates. In 1915, he deveoped an approximate method of soving the above probem and by doing it made an important and everasting contribution to mechanics. The theoretica foundation of the Gaerkin method goes back to the Principe of Virtua Work. We wi iustrate his idea on the exampe of the moderatey arge theory of beams. If we go back to Lecture 3 and foow the derivation of the equations of equiibrium from the variationa principe, the so caed weak form of the equiibrium is given by Eq. (3.54). Adding the non-inear term representing the contribution of finite rotations, this equation can be re-written as where (M + Nw + q)δw + N δu + Boundary terms (6.43) M = EIw N = EA[u + 1 2 (w ) 2 ] (6.44a) (6.44b) From the weak (goba) equiibrium one can derive the strong (oca) equiibrium by considering an infinite cass of variations. But, what happens if, instead of a cass, we consider ony one specific variation (shape) that satisfies kinematic boundary conditions? The equiibrium wi be vioated ocay, but can be satisfied gobay in average [ ( EIw IV + EA u + 1 )] 2 (w ) 2 w + q δw = (6.45) 6-8
Structura Mechanics 2.8 Lecture 6 Semester Yr Consider the exampe of a simpy supported beam, restrained from axia motion. The exact soution of this probem fro the sinusoida distribution of oad was given in the previous section. Assume now that the same beam is oaded by a uniform ine oad q(x) = q. No exact soution of this probem exists. Let s sove this probem approximatey by means of the Gaerkin method. As a tria approximate defected shape, we take the same shape that was found as a particuar soution of the fu equation w(x) = C sin πx δw(x) = δc sin πx (6.46a) (6.46b) With the condition of ends fixity in the axia direction, u = u =, and Eq. (6.45) yieds [ δc EIw IV + EA ] 2 (w ) 2 w + q sin πx = (6.47) Evauating the derivatives and integrating, the foowing expression is obtained 2 C + C 3 A q 1 2 8 2I EI( π )4 π = (6.48) is reated to the re- After re-arranging, the dimensioness defection ampitude C h = w o h maining w o h + 3 2 ( wo ) 3 ( q1 ) ( ) 48 4 = h Eh π 5 (6.49) h The above cubic equation has a simpe soution. Let s discuss the two imiting cases. Without the non-inear term, Eq. (6.49) predicts the foowing defection of the beam under pure bending action for the square section w ( o h = q1 ) 48 Eh π 5 ( h ) 4 (6.5) In the exact soution of the same probem, the numerica coefficient is 6 384 = 1, which is 6.4 ony 1.5% smaer than the present approximate soution 48 π 5 = 1. If on the other hand 6.3 the fexura resistance is sma, EI, the first term in Eq. (6.49) vanishes giving a cubic oad-defection reation ( wo ) 3 32 ( q1 ) ( ) 4 = h π 5 (6.51) Eh h There is no cosed-form soution for the pure membrane response of the beam under uniform pressure. However, the present prediction compares favoraby with the Eq. (6.4) for the moderatey arge defection, if the tota oad under the uniform and sinusoida pressure is the same P = q 1 = q o sin πx = q 2 o (6.52) π 6-9
Structura Mechanics 2.8 Lecture 6 Semester Yr Repacing q 1 by 2 π q o, the pure membrane soution takes the fina form ( wo ) 3 6.4 ( qo ) ( ) 4 = h π 4 (6.53) Eh h One can see that not ony the dimensioness form of the exact and approximate soutions are identica, but aso the coefficient 6.4 in Eq. (6.53) is of the same order as the coefficient 4 in Eq. (6.41). 6.5 Generaization to Arbitrary Non-inear Probems in Pates and Shes The previous section fet with the appication of the Gaerkin method to sove the non-inear ordinary differentia equation for the bending/membrane response of beams. Gaerkin name is forever attached to the anaytica or numerica soution of partia differentia equation, such as describing response of pates and shes. In the iterature you wi often encounter such expression as Gaerkin-Bubnov method, Petrov-Gaerkin method, the discontinuous Gaerkin method or the weighted residua method. The essence of this method is sketched beow. Denote by F (w, x) the non-inear operator (the eft hand side of the partia differentia equation) is defined over a certain fixed domain in the 2-D space S. Now, a distinction is made between the exact soution w (x) and the approximate soution w(x). The approximate soution is often referred to as a tria function. The exact soution makes the operator F to vanish F (w, x) = (6.54) The approximate soution does not satisfy exacty the governing equation, so instead of zero, there is a residue on the right hand of the Eq. (6.55) F (w, x) = R(x) (6.55) The residue can be positive over part of S and negative esewhere. If so, we can impose a weaker condition that the residue wi become zero in average over S, when mutipied by a weighting function w(x) R(x)w(x) ds = (6.56) S Mathematicay we say that these two functions are orthogona. In genera, there are aso boundary terms in the Gaerkin formuation. For exampe, in the theory of moderatey arge defection of pates, Eq. (6.56) takes the form (D 4 w N αβ w,αβ )w ds = (6.57) S 6-1
Structura Mechanics 2.8 Lecture 6 Semester Yr The counterpart of Eq. (6.57) in the theory of moderatey arge defection of beams is Eq. (6.47) which was soved in the previous section of the notes. The soution of partia differentia equations for both inear and non-inear probems is extensivey covered in textbooks on the finite eement method and therefore wi not be covered here. 6-11
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