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Static Analysis of a Variable Cross-Section Element Under Load

Static Analysis of a Variable Cross-Section Element Under Load Acta Sci. Pol. Architectura 20 (2) 2021, 27–40 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.2.12 ORIGINAL P APER Received: 28.12.2020 Accepted: 31.05.2021 STATIC ANALYSIS OF A VARIABLE CROSS-SECTION ELEMENT UNDER LOAD Jan Zamorowski Faculty of Materials, Civil and Environmental Engineering, University of Bielsko-Biala, Bielsko-Biała, Poland ABSTRACT An incremental solution of a member in bending and compression by displacement method with shear force impact is presented, wherein individual incremental steps, both the geometrical characteristics of the cross- -section and the position of the neutral axis, can be changed. Functions of loads, internal forces and displace- ments are represented by trigonometric series. In every load step internal forces and deformation state is memorised, therefore history of the load and stiffness is taken into account. This solution is useful both in static calculations of structures being strengthened as well as in structures, in which degradation of stiffness under load occurs as a result of wall stability loss. The member may be loaded by uniformly distributed load, any number of concentrated forces and nodal moments. Key words: incremental solution by displacement method, strengthening of structures, degradation of stiffness the member or at any node, it is necessary to consider INTRODUCTION such a possibility in the computer program algorithm Strengthening of steel bar structures is performed in by introducing into calculations the final configuration the presence of displacement and internal forces states, of nodes, rods and restraints like for non-deformed existing at the start of construction work. It may con- structure and the parameter controlling the rigidity of sist in enlargement of the member cross-section along the restraint and its initial deformation. For example, its entire length or its part, or introducing additional introducing an additional support of initially and elas- restraints in any cross-section or in structure nodes. tically deformed member would require introducing Therefore, to assess the effort of the strengthened mem- in any incremental step a movement of this additional ber, an incremental solution with possibility of chang- restraint relative to the non-deformed structure by the ing the stiffness of the member at every calculation sum of the values of initial and elastic member de- step is needed. Such an approach allows the analysis of formation. In addition, the current sum of initial and structures strengthened by symmetrical enlargement of elastic deflection of two members formed as a result the cross-section relative to the neutral axis. If change of supporting the original member, would have to be of the cross-section neutral axis position occurs, it is corrected. Similar operation would have to be car- still necessary to take into account the effect of nodal ried out in case of change in the cross-section of ana- bending moments resulting from the axial force and the lysed member along part of its length, with introduc- eccentric determined by the axis movement. tion of two additional nodes along the length of such In turn, if analysed structure is strengthened by a member, in the initial and in the end sections of the introducing additional restraints along the length of strengthened segment. Jan Zamorowski https://orcid.org/0000-0003-4394-4821 JZamorowski@ath.bielsko.pl © Copyright by Wydawnictwo SGGW Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 A similar solution would be needed in the event supplemented with the impact of node flexibility (Za- of degradation of the bar stiffness due to stability loss morowski, 2013). of its slender wall exposed to normal stress. However, due to the possible change in the value of normal stress BENDING IN THE xy PLANE along the bar length, such a solution should be treated as approximate as an effect of the need of splitting the Division of the load into any number of incremental bar into short elements, for which the stress values steps is introduced. In each incremental step, it is can be considered approximately constant along their possible to change the cross-section, assuming this lengths. In such elements, in subsequent load increase change will not affect the existing strain of the bar. steps, the surface area, moment of inertia and position Such a model of the bar allow the analysis of the of the neutral axis would change. structure with sequential application of loads (one There are solutions and calculation software, in after another), including alternating loads, as well as which the impact of local instability of slender walls on the analysis of erected or strengthened structure with the member global deformations is considered. In gen- variable topology. Two local coordinate systems were eral, these programs use shell models of bar (Dassault introduced – in the p and k node with positive senses Systèmes, 2013; ERGOCAD, 2020) and can be used as in Figure 1. to analyse individual thin-walled members or simple It was assumed that the positive senses of initial and frames (Giżejowski, Szczerba, Gajewski & Stachura, elastic strain of the bar and loads along its length are 2015). These models are accurate but their implemen- consistent with the positive senses of the system axis tation is time-consuming. In PhD dissertation (Cze- in the node p, whereas positive senses of nodal loads, piżak, 2006) the effect of trapezoidal sheets stiffness nodal forces and node displacements correspond to the degradation due to the local instability of their walls positive senses of the axes of the local systems starting is captured by function of vanishing stiffness r (M), at these nodes. describing the change in the stiffness of the corruga- It is assumed that in any incremental step k the ted sheet cross-section as a function of utilization its bending rigidity of the bar is equal to EJ , and shear z,k bending moment resistance. This solution can be used rigidity equal to GA . It is adopted that the bar de- in the first-order analysis of multi-span members sub- flection is the sum of initial bending (u ), deflection y,0 jected to bending without axial force. There are also caused by bending impact (u ) and deflection caused y,M solutions take into account degradation of the member by shear force (u ) – see Figure 2. y,V stiffness due to the development of the plastic zone as- sociated with the appearance of plastic hinges (Barszcz & Giżejowski, 2006). Unfortunately, these solutions are of little use to the engineer. They can only be used in individual cases to a limited extent. The article presents formulas used in displace- ment method in an incremental approach for ini- tially deformed bar, in compression and bending in the xy plane. These formulas may be used to assess the load capacity of members being strengthened, or whose cross-section is degraded due to loss of wall stability. Similar formulas can be derived for bend- ing in the xz plane, assuming the displacements and strains are low enough to make it possible to sum the stress caused by the impact of loads acting in both main planes, at the bar curvature determined from Fig. 1. Designation of load components, displacements two independent formulas.Such a solution can be still and nodal forces 28 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 2. Model of a to-pinned bar bent in the xy plane Total deflection in the first incremental step is in which b (y) and h = h + h – the section width and height 1 1 1,1 2,1 in the first step, Δ=uu +Δu +Δu (1) yy ,1 , 0 y,M,1 y,V, 1 A , A – cross-sectional area, shear cross-section- 1 v,1 al area in the first step, Increase in bending moment and shear force over J , S (y) – moment of inertia of the cross-section, z,1 z1 the length of the bar is determined by the following static moment of the cut-off part of the equations cross-section. The equation for the curvature of the bar in bend- Δ= Mx ΔM p+ΔN⋅Δu zz ,1 () ,1 ( ) 1 y,1 ing assuming small displacements is adopted in the (2a) du Δ y,1 following equation Δ= Vx () ΔV (p)+ΔN⋅ yy ,1 ,1 1 dx d Δu ΔM ( x ) y ,M ,1 z ,1 (4) ≅ = − where ρ ( x ) dx EJ y z ,1 By applying of double differentiation of Eq. (1) is Δ= Mp ΔM M +ΔM p + zz ,1 () ,1()z,p z,1()y obtained +ΔMP + ΔM M zy ,1() ,i z,1() zk , (2b) 2 2 2 2 d Δu d u d Δu d Δu y ,1 y , 0 y ,M ,1 y ,V ,1 (5) Δ= Vp () ΔV () M +ΔVp()+ yy ,1 ,1 z,p y,1 y = + + 2 2 2 2 dx dx dx dx +ΔVP + ΔV M yy ,1() ,i y,1 () zk , and, after introducing Eq. (2a) to Eq. (4) and taking Angle of shear deformation into account the derivative (5) and after ordering, it is obtained duΔΔV ()x yV,,1 y,1 == a y ,1 dx GA d Δu η y ,1 y ,1 (3) + k Δu = − ΔM ( p) + η y ,1 y ,1 z ,1 dx EJ du Δ z ,1 ªº y ,1 =Δ γ Vp()+ΔN yy ,1 ,1 1 «» dx ¬¼ d Δu dΔV ( p) y , 0 y ,1 (6) + η + η γ where (Belyayev, 1956) y ,1 y ,1 y ,1 dx dx 2,1 η N Sy() 1 y ,1 1 AA z ,1 for η = , N = ΔN , k = ad=≈y , y ,1 y ,1 1 1 y ,1 (1 − γ N ) EJ by() A J 1, v1 y ,1 1 z ,1 − h z ,1 1,1 architectura.actapol.net 29 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 In the second incremental step Δu = Δu + Δu (7) y , 2 y ,M , 2 y ,V , 2 ΔMx( ) = ΔM ()p +N Δu + ΔN() Δu +Δu = ΔM ()p +N Δu +ΔN Δu zz ,2 ,2 1y ,2 2 y ,1 y ,2 z ,2 2 y ,2 2 y ,1 (8a) for Δ= M (p )ΔMM ( )+ΔM (p )+ΔM (P )+ΔMM ( ) zz ,2 ,2 z ,p z ,2 y z ,2y ,i z ,2 zk , where Δ=MMΔ ()p+NΔe zp,,zp,2 2 y,2 (8b) Δ=MMΔ ()p+NΔe 2, y2 zk,, zk,2 wherein NN =Δ + ΔN , Δe – change of neutral axis position in the second step, 21 2 y,2 – increase in external nodal moments in nodes and k in the second step. ΔΔ Mp(), () M p zp,,2 zk,,2 Considering that duΔΔ du yy ,2 ,1 Δ= Vx() ΔV (p)+N +ΔN yy ,2 ,2 2 2 dx dx du Δ duΔΔ d u yV,,2 yy ,2 ,1 ªº =Δγγ V ()xV = (Δ+p)N +ΔN (9) yy ,2 ,2 y ,2 y ,2 2 2 «» dx dx dx ¬¼ γ = y ,2 GA v,2 in the second step the following equation is obtained 2 2 ªº duΔΔ η d V ()p dΔu yy ,2 ,2 y ,2 y ,1 (10) +Δ ku =− ΔM ()p +ΔN Δu +γη +ΔN yz ,2 [] ,2 2y ,1 y ,2y ,2«» 2 y ,2 2 2 EJ dx z ,2 dx dx «» ¬¼ η N y ,2 2 1 2 where η = and k = . y ,2 y ,2 1 − γ N EJ y ,2 2 z ,2 Similarly, for incremental step k (with k > 1), the following dependencies are obtained: Δ=uuΔ +Δu (11) yk,, y M,k y,V,k k −1 Δ= Mx () ΔM (p)+NΔu +ΔN Δu zk,,zk k y,k k y,m (12a) m=1 for (Δ= M p) ΔMM( )+ΔM (p)+ΔM (P )+ΔMM( ) zz ,2 ,2 z ,p z ,2 y z ,2y ,i z ,2 zk , k −1 Δ=MMΔ ()p+NΔe +ΔN Δe where zp,,zp k y,k k y,m m=2 (12b) k −1 Δ=MMΔ ()p+NΔe +ΔN Δe zk,,zk ky,, k k ym m=2 wherein NN =Δ ,Δe km y ,2 – change the position of the neutral axis in k-th step, m=1 ΔΔ Mp(), ( M p) – increase in external node moments in nodes p and k in the k-th step. zp,,k zk,,k Considering that k −1 duΔΔ d u yk,, y m (13) Δ= Vx () ΔV (p)+N +ΔN yk,, y k k k dx dx m=1 30 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 differential equation of deformed axis in incremental step k takes the following form 2 2 kk −− 11 ªº du Δ ηªº dV Δ ()p du Δ yk , yk,, yk ym , +Δ ku =− ΔM p+ΔN Δu +γη +ΔN yk,,z k () k y,m y,k y,k«» k «»¦¦ yk , 2 2 EJ dx zk , dx dx ¬¼mm == 11«» ¬¼ (14) h2 Sy() η N a yk , k yk , A zk , 1 2 k where: η== , k and γ ==ady . yk , yk,, yk yk , 1 − γ NEJ () GA b ( y) yk,, k z k kk − h zk , 1 The functions of initial bending of the bar (u ), elastic bending increments (Δu ), internal bending moments y,0 y,k increments (ΔM ) and load increments (Δp ) are assumed to be infinite trigonometric series z,k y,k nx π uf = sin yy ,0 ,0 ,n nx π Δ=uf sin yk,, yk,n (15) nx π Δ= Mp() Δm sin zk,, z k,n nx π sin Δ=ppΔ yk,, yk,n n = 1, 2, 3, ... The amplitudes of the increments of bending moments from external loads were obtained from the equation dV Δ ()p dM Δ ()p zk , yk , ==−Δp (16a) yk , dx dx in which Δp is the k-th increment of transverse load. y,k The following equation was obtained §· l Δ=mpΔ (16b) zk,,n y,, k n ¨¸ ©¹ nπ where (acc. to Girkmann, 1956) amplitude for an uniformly distributed load 4 Δ p yk , 2 nπ Δ= p sin yk,,n nπ 2 (16c) 4 Δpl yk , 2 nπ Δ= m sin zp,,k,n () nπ In case of concentrated load of simple-supported beam it can be written 2 Δ P yi,,k na π P i Δ= p sin yk,,n ll (16d) 2 ΔPl yi,,k na π Δ= m sin zP,,k,n () nπ where a is the distance of the force P from the node p. i y,i architectura.actapol.net 31 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 The series for concentrated force is not convergent, but for the bending moment series is. Assuming the concentrated force in the middle of the bar span, differences in the values of the maximum bending moment are about 1% with n = 40 and 0.5% with n = 80 and 0.25% with n = 160 compared to precise solution. Amplitudes of bending moments from loading with moments in the node p and k 2 ΔM zp,,k Δ= m zp,,k,n nπ (16e) 2 ΔM zk,,k Δ=mn − cos π zk,,k,n nπ In case of a beam loaded with bending moments in the support nodes, the differences between the value of bending moments obtained according to the first order theory and calculated from Eq. (15) with the amplitudes according to Eq. (16e), are larger in cross-sections close to the point of application of the load and smaller in the middle of the beam span. Assuming, for example, n = 1,000, these differences are: 2% in a cross-section at a distance of 0.01l from the node, 0.2% in cross-section at a distance of 0.1l and 0.06% in the middle of the bar span. Substituting e.g. Eqs. (16c) and (16e) to Eq. (15) for a uniformly distributed load and moments in nodes p and k it is obtained ªº 2 ΔM 42 ΔΔ pl M yk,,z p,k z,k,k 2nn ππx «» Δ= Mp ⋅ sin + − cosnπ sin (17) zk , () 2 nn ππ l «» n nπ () ¬¼ Assuming designations as in Eqs. (16c) and (16e) dVΔΔ ()p d M ()p yk,, zk §·nn ππx = = − ªº Δmm +Δ +Δm sin (18) zp,,k,n zp, ,k,n ¦ ¨¸ zk,,k,n ¬¼ dx ©¹ l l dx Substituting Eqs. (15), (17) and (18) and derivatives of displacements (15) to (6), the elastic curve equation describing deflection of the bar with the uniformly distributed load over the length and with moments in nodal cross-sections is obtained. In the first incremental step y ,1 §·nn ππx 2 nπx −+fk sin f sin=− ªº Δm +Δm +Δm × yn ,1, yn ,1, z ,p ,1,n z ,pn ,1, ¦¦ ¨¸ ¦ zk,,1,n y ,1 ¬¼ ©¹ll l EJ z ,1 nn n nxππ§·n nπx §·nπ nxπ × sin −ηηfm sin −γ ªº Δ +Δm +Δm sin yy ,1 ,0 ,n y ,1y ,1 z,p ,1,n z,p ,1,n ¦¦ ¨¸ ¨¸ zk,,1,n ¬¼ ll©¹ l ©¹l l nn (19) Hence, the bar bending amplitude after adding the influence of the concentrated force according to Eq. (16d), for any harmonic component n in the first incremental step is §· nπ η f yy ,1¨¸ ,0 ,n Δ+mmΔ +Δmm+Δ zp , ,1,n z,P,1,n z,p ,1,n zk,,1,n ©¹ l (20a) f=+ yn ,1, kr,, z1,n nπ §· 2 − N 1 − k ¨¸ y ,1 1 + γ N yk ,1 r,z,1,n ©¹ l nE π J z ,1 where N = . kr,, z 1,n 32 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Similarly, for the incremental step k: k −1 Δ+mmΔ +Δm +ΔN f z,, p kn, z,p,kn, k y,m,n zk,,k,n ¦ m=1 f = yk ,,n kr,, z k,n − N (20b) 1 + γ N yk,, kr z,k,n nE π J zk , N = kr,, z k,n k −1 where ym,,n is the sum of the amplitudes of elastic displacements from previous incremental steps. m=1 Substituting the amplitudes (20a) and (20b) in Eq. (15), it is possible to determine the elastic deflection of the bar Δu in any cross-section loaded with concentrated forces and the uniform load distributed over its length yk , and moments in node cross-sections. The increase in the elastic angle of rotation from bar bending in the first and in the k-th incremental steps is calculated from the following formulas (Fig. 3): du du du du du yy ,1 ,V ,1 y,0 y,1 y,0 φγ ()xV =− − = − Δ ()x− zy ,1 ,1y,1 dx dx dx dx dx (21) du du du yk,, yV,k y,k φγxV =− = − Δx () () z,, k yk yk, dx dx dx In the first step, for increase of evenly distributed load with increase in moments MM , is obtained zp,,1 zk,,1 duΔΔ d u du duΔ yy ,1 ªº ,1y ,0y ,1 ()xp−− ( ) = 1− γ − Δ=ϕγ ΔVN+Δ ()ΔN zy,,111 y, 1 y,11 «» dx dx dx dx ¬¼ (22a) du d Δu dM Δ ()p du yy ,0 ,1 y ,0 z ,1 −−γγΔΔ VN () p = () 1− −γ − yy ,1 , y ,1 y ,1 dx dx dx dx and, taking into account derivatives of displacements and moments nπ Δϕ ()x=−γªº Δm +Δ+mmm Δ+Δ × zy , ,1 z,p,k,n z,,P 1,n z,,p 1,n 1 ¦ zk,,1,n ¬¼ (22b) nxππn nπx nπ nxπ ×+ cos 1− γ ff cos − cos () ΔN yy 1, 1 1,n y,0,n ¦¦ ll l l l nn ab Fig. 3. Angular displacements of the cross-section near the joint in the xy plane: a – at the first step of increment; b – at the k-th step architectura.actapol.net 33 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 In the incremental step k: k −1 duΔΔªº dM Δ ()p du yk,, zk , ym ()xN 1−− γγ Δ= ϕzk,,() y k k y,k +ΔNk (22c) «» ¦ dx dx dx m=1 ¬¼ and after applying expressions for derivatives of displacements and moments nπ ()xm =−γªº Δ +Δ+m Δ+m Δm × Δϕzy , ,1 z,p,k,n z,,P 1,n z,,p 1,n 1 ¦ zk,,1,n ¬¼ (22d) nxππn nπx nπ nxπ ×+ cos 1− γ ff cos − cos ()yy 1, ΔN 1,n y,0,n 1¦¦ ll l l l nn In case of bar initially deformed in the xy plane, loaded with a bending moment in the node p as in Figure 4, the increase in the elastic angle of rotation in the first step is calculated according to Eq. (22b), taking into account the amplitude of the moment as in Eq. (16e). ªº 2 ΔM zp,,1 nn ππx ()xf=− 1γγ− −f cos (23) ΔΔ ϕzy , ,1,n()y ,1 N y ,1 y ,0 ,n ¦«» lnπ l n¬¼ Introducing the amplitude of the elastic bending deflection to Eq. (23), in the form resulting from Eq. (20a) for the moment load in the node p, assuming x = 0, an expression for the increase of the rotation angle in the first incremental step was obtained nN π f 1,yn0, 2 + ΔΔ MM zp,,1 z,p,1 Δϕ (24a) zp,,1 lN −+N (1 γ N ) kr,, z 1,n 1 y,1 kr,z,1,n Introducing the designation nN π f 1,yn0, 2 + 3EJ ΔM z,1 zp,,1 (24b) C = zp ,,1 NN−+ (1 γ N ) kr,, z n 1 y,1 kr, z,1,n in which the upper index describes the location of load application point, and the second lower index – the ro- tated node ΔMl zp,,1 pp = C (24c) Δϕ zp,, zp,,1 3EJ z ,1 Fig. 4. A bar with hinged ends loaded by the moment M zp , 34 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Similarly, for x = l ΔMl zp ,,1 pp Δϕ = C zk,, zk,,1 6 EJ z,1 (24d) nN π f 1,yn0, 2 + 6 EJ ΔM z ,1 zp,,1 for cos Cn = π zk,,1 NN−+ (1 γ N ) l kr,, z1,n 1 y,1 kr,z,1,n In incremental step k k −1 ªº 2 ΔM zp ,,k nn ππx ()xf=− 1γγN− −γ f cos (25a) ΔΔ ϕz,, k yk,n() y,k k yk, yk, Nk y,m,n «» ¦¦ lnπ l nm¬¼ =1 and ΔMl zp,,k pp = C Δϕ zp,,k z,,p k 3 EJ zk , (25b) ΔMl zp,,k pp = C Δϕ zk,,k z,k ,k 6 EJ zk , where k −1 nN π Δ f ky,,mn m=1 2 + 3EJ ΔM zk,, z p,k C = zp ,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n (25c) k −1 nN π Δ f ky,,mn m=1 2 + 6 EJ ΔM zk,,z p,k Cn = cos π zk,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n For a bar loaded with a moment in the end node as in Figure 5 the increase in the elastic angle of rotation in the first step is calculated according to Eq. (22b), after taking into account the amplitude of the moment as in Eq. (16e) and the displacement amplitude as in Eq. (20a). The following equation was obtained nN π f 1,yn0, − 2cos nπ ΔΔ MM (26a) zk,,1 zk, ,1 nx π () x = cos Δϕ z, 1 ¦ lN −+N (1 γ N ) l kr,, z1,n 1 y,1 kr,z,1,n and hence for x = 0 and for x = l ΔMl zk,,1 kk = C Δϕ zp,,1 z,p,1 6 EJ z ,1 (26b) ΔMl zk,,1 kk = C Δϕ zk,,1 z,k ,1 3EJ z ,1 architectura.actapol.net 35 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 5. A bar with hinged ends loaded by the moment M zk , wherein nN π f 1,yn0, − 2cos nπ ΔM 6 EJ zz ,1 ,k ,k C = zp ,,1 NN−+ (1 γ N ) kr,, z1,n 1 y,1 kr,z,1,n (26c) nN π f 1,yn0, − 2cos nπ ΔM 3EJ zz ,1 ,k ,k Cn = cos π zk,,1 2 NN−+ (1 γ N ) kr,, z1,n 1 y,1 kr,z,1,n In the incremental step k k −1 nN π Δ f ky,,mn m=1 (27a) − 2cos nπ ΔΔ MM zk,,k z,, k k nx π () x = cos Δϕ zk , lN −+N (1 γ N ) l kr,, z k,n k y,k kr,z,k,n and for x = 0 and x = l ΔMl zk,,k kk Δϕ = C zp,,k z,,p k 6 EJ zk , (27b) ΔMl zk,,k kk = C Δϕ zk,,k z,, k k 3EJ zk , where k −1 nN π Δ f ky,,mn m=1 − 2cos nπ ΔM 6 EJ zk,, zk,k C = zp ,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n (27c) k −1 nN π Δ f ky,,mn m=1 − 2cos nπ ΔM 3EJ zk,,zk,k Cn = cos π zk,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n In case of a initially deformed bar in compression, loaded with an uniformly distributed load of constant intensity as in Figure 6 the following is obtained 36 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ nn ππx Δϕ () x = cos (28a) zk , lN −− N γ N N l kr,,zk,n k y,kkkr,z,k,n k −1 where for k = 1 and for k > 1 MN =Δ f , hence, for x = 0 and for x = l MN = f ky,,mn 1,yn0, ¦ zk,,n zn ,1, m=1 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ q nπ Δϕ zp ,,k lN −− N γ N N kr,, z k,n k y,k k kr,z,k,n (28b) 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ q nπ = cos nπ Δϕ zk,,k lN −− N γ N N kr,, z k,n k y,k k kr,z,k,n Fig. 6. A bar with hinged ends loaded uniformly by the load p Similarly, it is possible to obtain transformation of columns fixed in foundations in both directions, formulas of the displacement method for bending in spaced longitudinally every 6.0 m. Eave beams sup- the xz plane. It should be taken into account that with ported on two columns are designed as a box sections the coordinate system as in Figure 1, the positive value composed of two C-profiles 240. These beams were of the node moment (with a vector compatible connected each other with tie rods with a diameter of yp , with the y axis) corresponds to the negative value of 24 mm located in planes of the columns and in planes the elastic bend deflection and the positive value of the located 3 m away from them. The covering was made initial bend deflection. of arched steel sections with cross-sectional dimen- sions as in Figure 7, with a thickness of 1 mm and Example a width of 600 mm between axes of their locks. The The solution presented earlier was supplemented with umbrella roof was located in the third wind load zone a module with rigidity degradation of thin-walled and the fourth snow load zone according to the Polish members exposed to normal stress, based on the National Annex to Eurocode 1 Part 4 (Polski Komitet EN 1993-1-3:2006 standard (European Committee for Normalizacyjny [PKN], 2010). Standardization [CEN], 2006).This extended solution The arch was treated as a system of pre-bent bar el- was used to analyse an individual arch of the umbrella ement in compression and bending, with compressive roof covering with characteristics as in Figure 7. The rigidity EA , bending rigidity EJ and shear rigidity k k supporting structure of the umbrella roof consists GA variable in subsequent incremental steps k. The v,k architectura.actapol.net 37 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 7. Characteristics of the umbrella roof value of the initial deflection was assumed consistent instability of the walls was evaluated at each incremen- with deflection of the arch with a chord equal to the tal step in a manner provided for in the PN-EN 1993- element length. In the solution, the effect of general -1-3:2008 standard (PKN, 2008). instability is considered as P – Δ effect, i.e. the effect Equations of displacement method derived as of changing the configuration of the bar members on shown before were used to build a computer program the values of nodal shear forces. The impact of local for calculating any planar bar systems (Fig. 8). Due to Fig. 8. Block diagram of the program for calculating arched trapezoidal corrugated sheets 38 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 capabilities of this software, two calculation schemes theory, which includes the possibility of changing of the umbrella roof transverse system were consid- the cross-section characteristics under the subsequent loads. This solution can be useful for both strength- ered. In the first scheme, considering eave beams non- ened structures and structures consisting of thin- -deformable (i.e. rigid), 1/10 stiffness of columns and -walled members, which walls lose their stability 1/5 stiffness of tie rods were assigned to the single arch sheet, which allowed to obtain reliable results for under increasing stress. arches in the column area. The second scheme takes In engineering practice, the utilisation index of the strengthened members in bending and compression is into account the additional impact of the eave beams calculated as the sum of the utilisation indexes of the deformability for corrugated arched sheets located in member before and after strengthening. For ex ample, the area of the largest deflections of these beams. The arch was divided into 14 elements. The results in case symmetrical cross-section strengthening, in obtained for the intermediate arch (according to the the first stage the ∆M (x) = ∆M (p) + ∆N · ∆u z,1 z,1 1 y,1 moment is considered and in the second stage the second scheme) are presented in Figure 9 regarding ∆M (x) = ∆M (p) + ∆N · ∆u . Compared with to unloaded arch. The solid line shows the results ob- z,2 z,2 2 y,2 Eq. (8a), the components ∆N · ∆u and ∆N · ∆u tained without the effect of local instability u , and the 1 1 y,2 2 y,1 broken line takes into account these instability u . The are thus omitted. In case of the load bearing capacity sum of the x coordinates of the elements nodes and of members check according to following equation their displacements u was treated as the abscissae of ΔΔ NM ΔN ΔM 1,Ed 1,Ed 2,Ed 2,Ed ++ + ≤ 1 (29) the graph, and the u displacements of nodes as the χχ NM N M 11,Rc 1,Rd 2 2,Rc 2,Rd ordinates. Differences in the values of vertical displacements the value of the bending moment calculated in the sec- of arch nodes in the key exceed slightly 5% of dis- ond stage on the basis of external loads should be in- placements of the arc calculated without the effect of creased by aforementioned omitted components. Then local instability of the walls. ∆M = ∆M (p) + ∆N · ∆u + ∆N · ∆u . 2,Ed 2 1 y,2 y,1 Effectiveness of strengthening the actual structure may be checked, for example, by means of innova- CONCLUSIONS tive strain and stress fiber Bragg grating (FBG) and The paper presents the incremental solution for a bar residual magnetic field (RMF) analyses (Juraszek, using the displacement method by the second order 2019, 2020). Fig. 9. Arc strains architectura.actapol.net 39 o blachownicowych elementach smukłościennych [Re- REFERENCES sistance assessment of steelplanarframefabricated from Barszcz, A. & Giżejowski, M. (2006). Analiza zaawanso- slender web plategirders]. Czasopismo Inżynierii Lą- wana w projektowaniu konstrukcji stalowych wg Euro- dowej, Środowiska i Architektury – Journal of Civil kodu 3. Inżynieria i Budownictwo, 9, 501–506. Engineering, Environment and Architecture, 62 (4/15), Belyayev, H. M. (1956). Soprotivleniye materialov. Mo- 73–92. skva: GIT-TL. Juraszek, J. (2019). Residual magnetic field for identifica- Czepiżak, D. (2006). Nośność graniczna lokalnie wzmoc- tion of damage in steel wire rope [Identyfikacja uszko- nionych wieloprzęsłowych blach fałdowych (PhD dis- dzeń lin stalowych za pomocą rezydualnego pola magne- sertation). Instytut Budownictwa Politechnika Wroc- tycznego]. Archives of Mining Sciences, 64 (1), 79–92. ławska, Wrocław. https://doi.org/10.24425/ams.2019.126273 Dassault Systèmes (2013). Abaqus 6.13 Online Documen- Juraszek, J. (2020). Fiber Bragg sensors on strain analysis tation. Providence, RI: Dassault Systèmes Simulia Cor- of power transmission lines. Materials, 13 (7), 1559. poration. Retrieved from: http://130.149.89.49:2080/ https://doi.org/10.3390/ma13071559 v6.13/index.html [access 02.04.2013]. Polski Komitet Normalizacyjny [PKN] (2008). Eurokod 3. ERGOCAD (2020). ConSteel 14 User Manual. Re- Projektowanie konstrukcji stalowych. Część 1–3: Regu- trieved from: https://www.consteelsoftware.eu/up- ły ogólne. Reguły uzupełniające dla konstrukcji z kształ- loads/9/9/7/7/99772912/consteel_14_user_manual [ac- towników i blach profilowanych na zimno (PN-EN 1993- cess 20.05.2020]. -1-3:2008). Warszawa: Polski Komitet Normalizacyjny. European Committee for Standardization [CEN] (2006). Polski Komitet Normalizacyjny [PKN] (2010). Eurokod 1. Eurocode 3. Design of steel structures. Part 1–3: Gener- Oddziaływania na konstrukcje. Część 1–4: Oddziaływania al rules. Supplementary rules for cold-formed members ogólne. Oddziaływania wiatru (PN-EN 1991-1-7:2008/ and sheeting (EN 1993-1-3:2006). Brussels: European /NA:2010). Warszawa: Polski Komitet Normalizacyjny. Committee for Standardization. Zamorowski, J. (2013). Przestrzenne konstrukcje prętowe Girkmann, K. (1956). Flächentragwerke. Wien: Springer- z geometrycznymi imperfekcjami i podatnymi węzłami -Verlag. [Spatial bar structures with geometrics imperfections Giżejowski, M., Szczerba, R., Gajewski, M. & Stachura, and flexible nodes]. Gliwice: Wydawnictwo Politechni- Z. (2015). Analiza nośności stalowej ramy płaskiej ki Śląskiej. STATYKA ELEMENTU ZE ZMIENNYM PRZEKROJEM POD OBCIĄŻENIEM STRESZCZENIE W artykule przedstawiono rozwiązanie przyrostowe ściskanego i zginanego pręta metodą przemieszczeń z uwzględnieniem siły poprzecznej, w którym w poszczególnych krokach przyrostowych można zmieniać charakterystyki geometryczne przekroju łącznie ze zmianą położenia osi obojętnej. Takie rozwiązanie jest przydatne zarówno w obliczeniach statycznych konstrukcji z prętami wzmacnianymi, jak i konstrukcji, w których następuje degradacja sztywności pod obciążeniem na skutek utraty stateczności ścianek. W roz- wiązaniu wykorzystano szeregi trygonometryczne do przedstawienia funkcji obciążeń, sił wewnętrznych i przemieszczeń. Do elementu mogą być przyłożone równomiernie rozłożone obciążenie oraz dowolna liczba sił skupionych i momentów przywęzłowych. Słowa kluczowe: rozwiązanie przyrostowe metodą przemieszczeń, konstrukcje wzmacniane, degradacja sztywności elementów http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Scientiarum Polonorum Architectura de Gruyter

Static Analysis of a Variable Cross-Section Element Under Load

Acta Scientiarum Polonorum Architectura , Volume 20 (2): 14 – Jun 1, 2021

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Acta Sci. Pol. Architectura 20 (2) 2021, 27–40 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.2.12 ORIGINAL P APER Received: 28.12.2020 Accepted: 31.05.2021 STATIC ANALYSIS OF A VARIABLE CROSS-SECTION ELEMENT UNDER LOAD Jan Zamorowski Faculty of Materials, Civil and Environmental Engineering, University of Bielsko-Biala, Bielsko-Biała, Poland ABSTRACT An incremental solution of a member in bending and compression by displacement method with shear force impact is presented, wherein individual incremental steps, both the geometrical characteristics of the cross- -section and the position of the neutral axis, can be changed. Functions of loads, internal forces and displace- ments are represented by trigonometric series. In every load step internal forces and deformation state is memorised, therefore history of the load and stiffness is taken into account. This solution is useful both in static calculations of structures being strengthened as well as in structures, in which degradation of stiffness under load occurs as a result of wall stability loss. The member may be loaded by uniformly distributed load, any number of concentrated forces and nodal moments. Key words: incremental solution by displacement method, strengthening of structures, degradation of stiffness the member or at any node, it is necessary to consider INTRODUCTION such a possibility in the computer program algorithm Strengthening of steel bar structures is performed in by introducing into calculations the final configuration the presence of displacement and internal forces states, of nodes, rods and restraints like for non-deformed existing at the start of construction work. It may con- structure and the parameter controlling the rigidity of sist in enlargement of the member cross-section along the restraint and its initial deformation. For example, its entire length or its part, or introducing additional introducing an additional support of initially and elas- restraints in any cross-section or in structure nodes. tically deformed member would require introducing Therefore, to assess the effort of the strengthened mem- in any incremental step a movement of this additional ber, an incremental solution with possibility of chang- restraint relative to the non-deformed structure by the ing the stiffness of the member at every calculation sum of the values of initial and elastic member de- step is needed. Such an approach allows the analysis of formation. In addition, the current sum of initial and structures strengthened by symmetrical enlargement of elastic deflection of two members formed as a result the cross-section relative to the neutral axis. If change of supporting the original member, would have to be of the cross-section neutral axis position occurs, it is corrected. Similar operation would have to be car- still necessary to take into account the effect of nodal ried out in case of change in the cross-section of ana- bending moments resulting from the axial force and the lysed member along part of its length, with introduc- eccentric determined by the axis movement. tion of two additional nodes along the length of such In turn, if analysed structure is strengthened by a member, in the initial and in the end sections of the introducing additional restraints along the length of strengthened segment. Jan Zamorowski https://orcid.org/0000-0003-4394-4821 JZamorowski@ath.bielsko.pl © Copyright by Wydawnictwo SGGW Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 A similar solution would be needed in the event supplemented with the impact of node flexibility (Za- of degradation of the bar stiffness due to stability loss morowski, 2013). of its slender wall exposed to normal stress. However, due to the possible change in the value of normal stress BENDING IN THE xy PLANE along the bar length, such a solution should be treated as approximate as an effect of the need of splitting the Division of the load into any number of incremental bar into short elements, for which the stress values steps is introduced. In each incremental step, it is can be considered approximately constant along their possible to change the cross-section, assuming this lengths. In such elements, in subsequent load increase change will not affect the existing strain of the bar. steps, the surface area, moment of inertia and position Such a model of the bar allow the analysis of the of the neutral axis would change. structure with sequential application of loads (one There are solutions and calculation software, in after another), including alternating loads, as well as which the impact of local instability of slender walls on the analysis of erected or strengthened structure with the member global deformations is considered. In gen- variable topology. Two local coordinate systems were eral, these programs use shell models of bar (Dassault introduced – in the p and k node with positive senses Systèmes, 2013; ERGOCAD, 2020) and can be used as in Figure 1. to analyse individual thin-walled members or simple It was assumed that the positive senses of initial and frames (Giżejowski, Szczerba, Gajewski & Stachura, elastic strain of the bar and loads along its length are 2015). These models are accurate but their implemen- consistent with the positive senses of the system axis tation is time-consuming. In PhD dissertation (Cze- in the node p, whereas positive senses of nodal loads, piżak, 2006) the effect of trapezoidal sheets stiffness nodal forces and node displacements correspond to the degradation due to the local instability of their walls positive senses of the axes of the local systems starting is captured by function of vanishing stiffness r (M), at these nodes. describing the change in the stiffness of the corruga- It is assumed that in any incremental step k the ted sheet cross-section as a function of utilization its bending rigidity of the bar is equal to EJ , and shear z,k bending moment resistance. This solution can be used rigidity equal to GA . It is adopted that the bar de- in the first-order analysis of multi-span members sub- flection is the sum of initial bending (u ), deflection y,0 jected to bending without axial force. There are also caused by bending impact (u ) and deflection caused y,M solutions take into account degradation of the member by shear force (u ) – see Figure 2. y,V stiffness due to the development of the plastic zone as- sociated with the appearance of plastic hinges (Barszcz & Giżejowski, 2006). Unfortunately, these solutions are of little use to the engineer. They can only be used in individual cases to a limited extent. The article presents formulas used in displace- ment method in an incremental approach for ini- tially deformed bar, in compression and bending in the xy plane. These formulas may be used to assess the load capacity of members being strengthened, or whose cross-section is degraded due to loss of wall stability. Similar formulas can be derived for bend- ing in the xz plane, assuming the displacements and strains are low enough to make it possible to sum the stress caused by the impact of loads acting in both main planes, at the bar curvature determined from Fig. 1. Designation of load components, displacements two independent formulas.Such a solution can be still and nodal forces 28 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 2. Model of a to-pinned bar bent in the xy plane Total deflection in the first incremental step is in which b (y) and h = h + h – the section width and height 1 1 1,1 2,1 in the first step, Δ=uu +Δu +Δu (1) yy ,1 , 0 y,M,1 y,V, 1 A , A – cross-sectional area, shear cross-section- 1 v,1 al area in the first step, Increase in bending moment and shear force over J , S (y) – moment of inertia of the cross-section, z,1 z1 the length of the bar is determined by the following static moment of the cut-off part of the equations cross-section. The equation for the curvature of the bar in bend- Δ= Mx ΔM p+ΔN⋅Δu zz ,1 () ,1 ( ) 1 y,1 ing assuming small displacements is adopted in the (2a) du Δ y,1 following equation Δ= Vx () ΔV (p)+ΔN⋅ yy ,1 ,1 1 dx d Δu ΔM ( x ) y ,M ,1 z ,1 (4) ≅ = − where ρ ( x ) dx EJ y z ,1 By applying of double differentiation of Eq. (1) is Δ= Mp ΔM M +ΔM p + zz ,1 () ,1()z,p z,1()y obtained +ΔMP + ΔM M zy ,1() ,i z,1() zk , (2b) 2 2 2 2 d Δu d u d Δu d Δu y ,1 y , 0 y ,M ,1 y ,V ,1 (5) Δ= Vp () ΔV () M +ΔVp()+ yy ,1 ,1 z,p y,1 y = + + 2 2 2 2 dx dx dx dx +ΔVP + ΔV M yy ,1() ,i y,1 () zk , and, after introducing Eq. (2a) to Eq. (4) and taking Angle of shear deformation into account the derivative (5) and after ordering, it is obtained duΔΔV ()x yV,,1 y,1 == a y ,1 dx GA d Δu η y ,1 y ,1 (3) + k Δu = − ΔM ( p) + η y ,1 y ,1 z ,1 dx EJ du Δ z ,1 ªº y ,1 =Δ γ Vp()+ΔN yy ,1 ,1 1 «» dx ¬¼ d Δu dΔV ( p) y , 0 y ,1 (6) + η + η γ where (Belyayev, 1956) y ,1 y ,1 y ,1 dx dx 2,1 η N Sy() 1 y ,1 1 AA z ,1 for η = , N = ΔN , k = ad=≈y , y ,1 y ,1 1 1 y ,1 (1 − γ N ) EJ by() A J 1, v1 y ,1 1 z ,1 − h z ,1 1,1 architectura.actapol.net 29 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 In the second incremental step Δu = Δu + Δu (7) y , 2 y ,M , 2 y ,V , 2 ΔMx( ) = ΔM ()p +N Δu + ΔN() Δu +Δu = ΔM ()p +N Δu +ΔN Δu zz ,2 ,2 1y ,2 2 y ,1 y ,2 z ,2 2 y ,2 2 y ,1 (8a) for Δ= M (p )ΔMM ( )+ΔM (p )+ΔM (P )+ΔMM ( ) zz ,2 ,2 z ,p z ,2 y z ,2y ,i z ,2 zk , where Δ=MMΔ ()p+NΔe zp,,zp,2 2 y,2 (8b) Δ=MMΔ ()p+NΔe 2, y2 zk,, zk,2 wherein NN =Δ + ΔN , Δe – change of neutral axis position in the second step, 21 2 y,2 – increase in external nodal moments in nodes and k in the second step. ΔΔ Mp(), () M p zp,,2 zk,,2 Considering that duΔΔ du yy ,2 ,1 Δ= Vx() ΔV (p)+N +ΔN yy ,2 ,2 2 2 dx dx du Δ duΔΔ d u yV,,2 yy ,2 ,1 ªº =Δγγ V ()xV = (Δ+p)N +ΔN (9) yy ,2 ,2 y ,2 y ,2 2 2 «» dx dx dx ¬¼ γ = y ,2 GA v,2 in the second step the following equation is obtained 2 2 ªº duΔΔ η d V ()p dΔu yy ,2 ,2 y ,2 y ,1 (10) +Δ ku =− ΔM ()p +ΔN Δu +γη +ΔN yz ,2 [] ,2 2y ,1 y ,2y ,2«» 2 y ,2 2 2 EJ dx z ,2 dx dx «» ¬¼ η N y ,2 2 1 2 where η = and k = . y ,2 y ,2 1 − γ N EJ y ,2 2 z ,2 Similarly, for incremental step k (with k > 1), the following dependencies are obtained: Δ=uuΔ +Δu (11) yk,, y M,k y,V,k k −1 Δ= Mx () ΔM (p)+NΔu +ΔN Δu zk,,zk k y,k k y,m (12a) m=1 for (Δ= M p) ΔMM( )+ΔM (p)+ΔM (P )+ΔMM( ) zz ,2 ,2 z ,p z ,2 y z ,2y ,i z ,2 zk , k −1 Δ=MMΔ ()p+NΔe +ΔN Δe where zp,,zp k y,k k y,m m=2 (12b) k −1 Δ=MMΔ ()p+NΔe +ΔN Δe zk,,zk ky,, k k ym m=2 wherein NN =Δ ,Δe km y ,2 – change the position of the neutral axis in k-th step, m=1 ΔΔ Mp(), ( M p) – increase in external node moments in nodes p and k in the k-th step. zp,,k zk,,k Considering that k −1 duΔΔ d u yk,, y m (13) Δ= Vx () ΔV (p)+N +ΔN yk,, y k k k dx dx m=1 30 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 differential equation of deformed axis in incremental step k takes the following form 2 2 kk −− 11 ªº du Δ ηªº dV Δ ()p du Δ yk , yk,, yk ym , +Δ ku =− ΔM p+ΔN Δu +γη +ΔN yk,,z k () k y,m y,k y,k«» k «»¦¦ yk , 2 2 EJ dx zk , dx dx ¬¼mm == 11«» ¬¼ (14) h2 Sy() η N a yk , k yk , A zk , 1 2 k where: η== , k and γ ==ady . yk , yk,, yk yk , 1 − γ NEJ () GA b ( y) yk,, k z k kk − h zk , 1 The functions of initial bending of the bar (u ), elastic bending increments (Δu ), internal bending moments y,0 y,k increments (ΔM ) and load increments (Δp ) are assumed to be infinite trigonometric series z,k y,k nx π uf = sin yy ,0 ,0 ,n nx π Δ=uf sin yk,, yk,n (15) nx π Δ= Mp() Δm sin zk,, z k,n nx π sin Δ=ppΔ yk,, yk,n n = 1, 2, 3, ... The amplitudes of the increments of bending moments from external loads were obtained from the equation dV Δ ()p dM Δ ()p zk , yk , ==−Δp (16a) yk , dx dx in which Δp is the k-th increment of transverse load. y,k The following equation was obtained §· l Δ=mpΔ (16b) zk,,n y,, k n ¨¸ ©¹ nπ where (acc. to Girkmann, 1956) amplitude for an uniformly distributed load 4 Δ p yk , 2 nπ Δ= p sin yk,,n nπ 2 (16c) 4 Δpl yk , 2 nπ Δ= m sin zp,,k,n () nπ In case of concentrated load of simple-supported beam it can be written 2 Δ P yi,,k na π P i Δ= p sin yk,,n ll (16d) 2 ΔPl yi,,k na π Δ= m sin zP,,k,n () nπ where a is the distance of the force P from the node p. i y,i architectura.actapol.net 31 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 The series for concentrated force is not convergent, but for the bending moment series is. Assuming the concentrated force in the middle of the bar span, differences in the values of the maximum bending moment are about 1% with n = 40 and 0.5% with n = 80 and 0.25% with n = 160 compared to precise solution. Amplitudes of bending moments from loading with moments in the node p and k 2 ΔM zp,,k Δ= m zp,,k,n nπ (16e) 2 ΔM zk,,k Δ=mn − cos π zk,,k,n nπ In case of a beam loaded with bending moments in the support nodes, the differences between the value of bending moments obtained according to the first order theory and calculated from Eq. (15) with the amplitudes according to Eq. (16e), are larger in cross-sections close to the point of application of the load and smaller in the middle of the beam span. Assuming, for example, n = 1,000, these differences are: 2% in a cross-section at a distance of 0.01l from the node, 0.2% in cross-section at a distance of 0.1l and 0.06% in the middle of the bar span. Substituting e.g. Eqs. (16c) and (16e) to Eq. (15) for a uniformly distributed load and moments in nodes p and k it is obtained ªº 2 ΔM 42 ΔΔ pl M yk,,z p,k z,k,k 2nn ππx «» Δ= Mp ⋅ sin + − cosnπ sin (17) zk , () 2 nn ππ l «» n nπ () ¬¼ Assuming designations as in Eqs. (16c) and (16e) dVΔΔ ()p d M ()p yk,, zk §·nn ππx = = − ªº Δmm +Δ +Δm sin (18) zp,,k,n zp, ,k,n ¦ ¨¸ zk,,k,n ¬¼ dx ©¹ l l dx Substituting Eqs. (15), (17) and (18) and derivatives of displacements (15) to (6), the elastic curve equation describing deflection of the bar with the uniformly distributed load over the length and with moments in nodal cross-sections is obtained. In the first incremental step y ,1 §·nn ππx 2 nπx −+fk sin f sin=− ªº Δm +Δm +Δm × yn ,1, yn ,1, z ,p ,1,n z ,pn ,1, ¦¦ ¨¸ ¦ zk,,1,n y ,1 ¬¼ ©¹ll l EJ z ,1 nn n nxππ§·n nπx §·nπ nxπ × sin −ηηfm sin −γ ªº Δ +Δm +Δm sin yy ,1 ,0 ,n y ,1y ,1 z,p ,1,n z,p ,1,n ¦¦ ¨¸ ¨¸ zk,,1,n ¬¼ ll©¹ l ©¹l l nn (19) Hence, the bar bending amplitude after adding the influence of the concentrated force according to Eq. (16d), for any harmonic component n in the first incremental step is §· nπ η f yy ,1¨¸ ,0 ,n Δ+mmΔ +Δmm+Δ zp , ,1,n z,P,1,n z,p ,1,n zk,,1,n ©¹ l (20a) f=+ yn ,1, kr,, z1,n nπ §· 2 − N 1 − k ¨¸ y ,1 1 + γ N yk ,1 r,z,1,n ©¹ l nE π J z ,1 where N = . kr,, z 1,n 32 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Similarly, for the incremental step k: k −1 Δ+mmΔ +Δm +ΔN f z,, p kn, z,p,kn, k y,m,n zk,,k,n ¦ m=1 f = yk ,,n kr,, z k,n − N (20b) 1 + γ N yk,, kr z,k,n nE π J zk , N = kr,, z k,n k −1 where ym,,n is the sum of the amplitudes of elastic displacements from previous incremental steps. m=1 Substituting the amplitudes (20a) and (20b) in Eq. (15), it is possible to determine the elastic deflection of the bar Δu in any cross-section loaded with concentrated forces and the uniform load distributed over its length yk , and moments in node cross-sections. The increase in the elastic angle of rotation from bar bending in the first and in the k-th incremental steps is calculated from the following formulas (Fig. 3): du du du du du yy ,1 ,V ,1 y,0 y,1 y,0 φγ ()xV =− − = − Δ ()x− zy ,1 ,1y,1 dx dx dx dx dx (21) du du du yk,, yV,k y,k φγxV =− = − Δx () () z,, k yk yk, dx dx dx In the first step, for increase of evenly distributed load with increase in moments MM , is obtained zp,,1 zk,,1 duΔΔ d u du duΔ yy ,1 ªº ,1y ,0y ,1 ()xp−− ( ) = 1− γ − Δ=ϕγ ΔVN+Δ ()ΔN zy,,111 y, 1 y,11 «» dx dx dx dx ¬¼ (22a) du d Δu dM Δ ()p du yy ,0 ,1 y ,0 z ,1 −−γγΔΔ VN () p = () 1− −γ − yy ,1 , y ,1 y ,1 dx dx dx dx and, taking into account derivatives of displacements and moments nπ Δϕ ()x=−γªº Δm +Δ+mmm Δ+Δ × zy , ,1 z,p,k,n z,,P 1,n z,,p 1,n 1 ¦ zk,,1,n ¬¼ (22b) nxππn nπx nπ nxπ ×+ cos 1− γ ff cos − cos () ΔN yy 1, 1 1,n y,0,n ¦¦ ll l l l nn ab Fig. 3. Angular displacements of the cross-section near the joint in the xy plane: a – at the first step of increment; b – at the k-th step architectura.actapol.net 33 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 In the incremental step k: k −1 duΔΔªº dM Δ ()p du yk,, zk , ym ()xN 1−− γγ Δ= ϕzk,,() y k k y,k +ΔNk (22c) «» ¦ dx dx dx m=1 ¬¼ and after applying expressions for derivatives of displacements and moments nπ ()xm =−γªº Δ +Δ+m Δ+m Δm × Δϕzy , ,1 z,p,k,n z,,P 1,n z,,p 1,n 1 ¦ zk,,1,n ¬¼ (22d) nxππn nπx nπ nxπ ×+ cos 1− γ ff cos − cos ()yy 1, ΔN 1,n y,0,n 1¦¦ ll l l l nn In case of bar initially deformed in the xy plane, loaded with a bending moment in the node p as in Figure 4, the increase in the elastic angle of rotation in the first step is calculated according to Eq. (22b), taking into account the amplitude of the moment as in Eq. (16e). ªº 2 ΔM zp,,1 nn ππx ()xf=− 1γγ− −f cos (23) ΔΔ ϕzy , ,1,n()y ,1 N y ,1 y ,0 ,n ¦«» lnπ l n¬¼ Introducing the amplitude of the elastic bending deflection to Eq. (23), in the form resulting from Eq. (20a) for the moment load in the node p, assuming x = 0, an expression for the increase of the rotation angle in the first incremental step was obtained nN π f 1,yn0, 2 + ΔΔ MM zp,,1 z,p,1 Δϕ (24a) zp,,1 lN −+N (1 γ N ) kr,, z 1,n 1 y,1 kr,z,1,n Introducing the designation nN π f 1,yn0, 2 + 3EJ ΔM z,1 zp,,1 (24b) C = zp ,,1 NN−+ (1 γ N ) kr,, z n 1 y,1 kr, z,1,n in which the upper index describes the location of load application point, and the second lower index – the ro- tated node ΔMl zp,,1 pp = C (24c) Δϕ zp,, zp,,1 3EJ z ,1 Fig. 4. A bar with hinged ends loaded by the moment M zp , 34 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Similarly, for x = l ΔMl zp ,,1 pp Δϕ = C zk,, zk,,1 6 EJ z,1 (24d) nN π f 1,yn0, 2 + 6 EJ ΔM z ,1 zp,,1 for cos Cn = π zk,,1 NN−+ (1 γ N ) l kr,, z1,n 1 y,1 kr,z,1,n In incremental step k k −1 ªº 2 ΔM zp ,,k nn ππx ()xf=− 1γγN− −γ f cos (25a) ΔΔ ϕz,, k yk,n() y,k k yk, yk, Nk y,m,n «» ¦¦ lnπ l nm¬¼ =1 and ΔMl zp,,k pp = C Δϕ zp,,k z,,p k 3 EJ zk , (25b) ΔMl zp,,k pp = C Δϕ zk,,k z,k ,k 6 EJ zk , where k −1 nN π Δ f ky,,mn m=1 2 + 3EJ ΔM zk,, z p,k C = zp ,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n (25c) k −1 nN π Δ f ky,,mn m=1 2 + 6 EJ ΔM zk,,z p,k Cn = cos π zk,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n For a bar loaded with a moment in the end node as in Figure 5 the increase in the elastic angle of rotation in the first step is calculated according to Eq. (22b), after taking into account the amplitude of the moment as in Eq. (16e) and the displacement amplitude as in Eq. (20a). The following equation was obtained nN π f 1,yn0, − 2cos nπ ΔΔ MM (26a) zk,,1 zk, ,1 nx π () x = cos Δϕ z, 1 ¦ lN −+N (1 γ N ) l kr,, z1,n 1 y,1 kr,z,1,n and hence for x = 0 and for x = l ΔMl zk,,1 kk = C Δϕ zp,,1 z,p,1 6 EJ z ,1 (26b) ΔMl zk,,1 kk = C Δϕ zk,,1 z,k ,1 3EJ z ,1 architectura.actapol.net 35 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 5. A bar with hinged ends loaded by the moment M zk , wherein nN π f 1,yn0, − 2cos nπ ΔM 6 EJ zz ,1 ,k ,k C = zp ,,1 NN−+ (1 γ N ) kr,, z1,n 1 y,1 kr,z,1,n (26c) nN π f 1,yn0, − 2cos nπ ΔM 3EJ zz ,1 ,k ,k Cn = cos π zk,,1 2 NN−+ (1 γ N ) kr,, z1,n 1 y,1 kr,z,1,n In the incremental step k k −1 nN π Δ f ky,,mn m=1 (27a) − 2cos nπ ΔΔ MM zk,,k z,, k k nx π () x = cos Δϕ zk , lN −+N (1 γ N ) l kr,, z k,n k y,k kr,z,k,n and for x = 0 and x = l ΔMl zk,,k kk Δϕ = C zp,,k z,,p k 6 EJ zk , (27b) ΔMl zk,,k kk = C Δϕ zk,,k z,, k k 3EJ zk , where k −1 nN π Δ f ky,,mn m=1 − 2cos nπ ΔM 6 EJ zk,, zk,k C = zp ,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n (27c) k −1 nN π Δ f ky,,mn m=1 − 2cos nπ ΔM 3EJ zk,,zk,k Cn = cos π zk,,k NN−+ (1 γ N ) kr,, z k,n k y,k kr,, z k,n In case of a initially deformed bar in compression, loaded with an uniformly distributed load of constant intensity as in Figure 6 the following is obtained 36 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ nn ππx Δϕ () x = cos (28a) zk , lN −− N γ N N l kr,,zk,n k y,kkkr,z,k,n k −1 where for k = 1 and for k > 1 MN =Δ f , hence, for x = 0 and for x = l MN = f ky,,mn 1,yn0, ¦ zk,,n zn ,1, m=1 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ q nπ Δϕ zp ,,k lN −− N γ N N kr,, z k,n k y,k k kr,z,k,n (28b) 4 Δpl yk , 2 nπ N sin + M zk ,,n () nπ q nπ = cos nπ Δϕ zk,,k lN −− N γ N N kr,, z k,n k y,k k kr,z,k,n Fig. 6. A bar with hinged ends loaded uniformly by the load p Similarly, it is possible to obtain transformation of columns fixed in foundations in both directions, formulas of the displacement method for bending in spaced longitudinally every 6.0 m. Eave beams sup- the xz plane. It should be taken into account that with ported on two columns are designed as a box sections the coordinate system as in Figure 1, the positive value composed of two C-profiles 240. These beams were of the node moment (with a vector compatible connected each other with tie rods with a diameter of yp , with the y axis) corresponds to the negative value of 24 mm located in planes of the columns and in planes the elastic bend deflection and the positive value of the located 3 m away from them. The covering was made initial bend deflection. of arched steel sections with cross-sectional dimen- sions as in Figure 7, with a thickness of 1 mm and Example a width of 600 mm between axes of their locks. The The solution presented earlier was supplemented with umbrella roof was located in the third wind load zone a module with rigidity degradation of thin-walled and the fourth snow load zone according to the Polish members exposed to normal stress, based on the National Annex to Eurocode 1 Part 4 (Polski Komitet EN 1993-1-3:2006 standard (European Committee for Normalizacyjny [PKN], 2010). Standardization [CEN], 2006).This extended solution The arch was treated as a system of pre-bent bar el- was used to analyse an individual arch of the umbrella ement in compression and bending, with compressive roof covering with characteristics as in Figure 7. The rigidity EA , bending rigidity EJ and shear rigidity k k supporting structure of the umbrella roof consists GA variable in subsequent incremental steps k. The v,k architectura.actapol.net 37 Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 Fig. 7. Characteristics of the umbrella roof value of the initial deflection was assumed consistent instability of the walls was evaluated at each incremen- with deflection of the arch with a chord equal to the tal step in a manner provided for in the PN-EN 1993- element length. In the solution, the effect of general -1-3:2008 standard (PKN, 2008). instability is considered as P – Δ effect, i.e. the effect Equations of displacement method derived as of changing the configuration of the bar members on shown before were used to build a computer program the values of nodal shear forces. The impact of local for calculating any planar bar systems (Fig. 8). Due to Fig. 8. Block diagram of the program for calculating arched trapezoidal corrugated sheets 38 architectura.actapol.net Zamorowski, J. (2021). Static analysis of a variable cross-section element under load. Acta Sci. Pol. Architectura, 20 (2), 27–40. doi: 10.22630/ASPA.2021.20.2.12 capabilities of this software, two calculation schemes theory, which includes the possibility of changing of the umbrella roof transverse system were consid- the cross-section characteristics under the subsequent loads. This solution can be useful for both strength- ered. In the first scheme, considering eave beams non- ened structures and structures consisting of thin- -deformable (i.e. rigid), 1/10 stiffness of columns and -walled members, which walls lose their stability 1/5 stiffness of tie rods were assigned to the single arch sheet, which allowed to obtain reliable results for under increasing stress. arches in the column area. The second scheme takes In engineering practice, the utilisation index of the strengthened members in bending and compression is into account the additional impact of the eave beams calculated as the sum of the utilisation indexes of the deformability for corrugated arched sheets located in member before and after strengthening. For ex ample, the area of the largest deflections of these beams. The arch was divided into 14 elements. The results in case symmetrical cross-section strengthening, in obtained for the intermediate arch (according to the the first stage the ∆M (x) = ∆M (p) + ∆N · ∆u z,1 z,1 1 y,1 moment is considered and in the second stage the second scheme) are presented in Figure 9 regarding ∆M (x) = ∆M (p) + ∆N · ∆u . Compared with to unloaded arch. The solid line shows the results ob- z,2 z,2 2 y,2 Eq. (8a), the components ∆N · ∆u and ∆N · ∆u tained without the effect of local instability u , and the 1 1 y,2 2 y,1 broken line takes into account these instability u . The are thus omitted. In case of the load bearing capacity sum of the x coordinates of the elements nodes and of members check according to following equation their displacements u was treated as the abscissae of ΔΔ NM ΔN ΔM 1,Ed 1,Ed 2,Ed 2,Ed ++ + ≤ 1 (29) the graph, and the u displacements of nodes as the χχ NM N M 11,Rc 1,Rd 2 2,Rc 2,Rd ordinates. Differences in the values of vertical displacements the value of the bending moment calculated in the sec- of arch nodes in the key exceed slightly 5% of dis- ond stage on the basis of external loads should be in- placements of the arc calculated without the effect of creased by aforementioned omitted components. Then local instability of the walls. ∆M = ∆M (p) + ∆N · ∆u + ∆N · ∆u . 2,Ed 2 1 y,2 y,1 Effectiveness of strengthening the actual structure may be checked, for example, by means of innova- CONCLUSIONS tive strain and stress fiber Bragg grating (FBG) and The paper presents the incremental solution for a bar residual magnetic field (RMF) analyses (Juraszek, using the displacement method by the second order 2019, 2020). Fig. 9. Arc strains architectura.actapol.net 39 o blachownicowych elementach smukłościennych [Re- REFERENCES sistance assessment of steelplanarframefabricated from Barszcz, A. & Giżejowski, M. (2006). Analiza zaawanso- slender web plategirders]. Czasopismo Inżynierii Lą- wana w projektowaniu konstrukcji stalowych wg Euro- dowej, Środowiska i Architektury – Journal of Civil kodu 3. Inżynieria i Budownictwo, 9, 501–506. Engineering, Environment and Architecture, 62 (4/15), Belyayev, H. M. (1956). Soprotivleniye materialov. Mo- 73–92. skva: GIT-TL. Juraszek, J. (2019). Residual magnetic field for identifica- Czepiżak, D. (2006). Nośność graniczna lokalnie wzmoc- tion of damage in steel wire rope [Identyfikacja uszko- nionych wieloprzęsłowych blach fałdowych (PhD dis- dzeń lin stalowych za pomocą rezydualnego pola magne- sertation). Instytut Budownictwa Politechnika Wroc- tycznego]. Archives of Mining Sciences, 64 (1), 79–92. ławska, Wrocław. https://doi.org/10.24425/ams.2019.126273 Dassault Systèmes (2013). Abaqus 6.13 Online Documen- Juraszek, J. (2020). Fiber Bragg sensors on strain analysis tation. Providence, RI: Dassault Systèmes Simulia Cor- of power transmission lines. Materials, 13 (7), 1559. poration. Retrieved from: http://130.149.89.49:2080/ https://doi.org/10.3390/ma13071559 v6.13/index.html [access 02.04.2013]. Polski Komitet Normalizacyjny [PKN] (2008). Eurokod 3. ERGOCAD (2020). ConSteel 14 User Manual. Re- Projektowanie konstrukcji stalowych. Część 1–3: Regu- trieved from: https://www.consteelsoftware.eu/up- ły ogólne. Reguły uzupełniające dla konstrukcji z kształ- loads/9/9/7/7/99772912/consteel_14_user_manual [ac- towników i blach profilowanych na zimno (PN-EN 1993- cess 20.05.2020]. -1-3:2008). Warszawa: Polski Komitet Normalizacyjny. European Committee for Standardization [CEN] (2006). Polski Komitet Normalizacyjny [PKN] (2010). Eurokod 1. Eurocode 3. Design of steel structures. Part 1–3: Gener- Oddziaływania na konstrukcje. Część 1–4: Oddziaływania al rules. Supplementary rules for cold-formed members ogólne. Oddziaływania wiatru (PN-EN 1991-1-7:2008/ and sheeting (EN 1993-1-3:2006). Brussels: European /NA:2010). Warszawa: Polski Komitet Normalizacyjny. Committee for Standardization. Zamorowski, J. (2013). Przestrzenne konstrukcje prętowe Girkmann, K. (1956). Flächentragwerke. Wien: Springer- z geometrycznymi imperfekcjami i podatnymi węzłami -Verlag. [Spatial bar structures with geometrics imperfections Giżejowski, M., Szczerba, R., Gajewski, M. & Stachura, and flexible nodes]. Gliwice: Wydawnictwo Politechni- Z. (2015). Analiza nośności stalowej ramy płaskiej ki Śląskiej. STATYKA ELEMENTU ZE ZMIENNYM PRZEKROJEM POD OBCIĄŻENIEM STRESZCZENIE W artykule przedstawiono rozwiązanie przyrostowe ściskanego i zginanego pręta metodą przemieszczeń z uwzględnieniem siły poprzecznej, w którym w poszczególnych krokach przyrostowych można zmieniać charakterystyki geometryczne przekroju łącznie ze zmianą położenia osi obojętnej. Takie rozwiązanie jest przydatne zarówno w obliczeniach statycznych konstrukcji z prętami wzmacnianymi, jak i konstrukcji, w których następuje degradacja sztywności pod obciążeniem na skutek utraty stateczności ścianek. W roz- wiązaniu wykorzystano szeregi trygonometryczne do przedstawienia funkcji obciążeń, sił wewnętrznych i przemieszczeń. Do elementu mogą być przyłożone równomiernie rozłożone obciążenie oraz dowolna liczba sił skupionych i momentów przywęzłowych. Słowa kluczowe: rozwiązanie przyrostowe metodą przemieszczeń, konstrukcje wzmacniane, degradacja sztywności elementów

Journal

Acta Scientiarum Polonorum Architecturade Gruyter

Published: Jun 1, 2021

Keywords: incremental solution by displacement method; strengthening of structures; degradation of stiffness

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