纖維增強(qiáng)聚合物布加固黏彈性木梁彎曲的解析解

歐陽煜,江勇,周磊

(上海大學(xué)土木工程系,上海200444)

纖維增強(qiáng)聚合物布加固黏彈性木梁彎曲的解析解

歐陽煜,江勇,周磊

(上海大學(xué)土木工程系,上海200444)

將木梁視為服從標(biāo)準(zhǔn)線性固體本構(gòu)的黏彈性體,假定纖維增強(qiáng)聚合物(fiber reinforced polymer,FRP)布與木梁緊密粘貼,研究了FRP布加固木梁線性彎曲的蠕變行為.在建立FRP布加固黏彈性矩形截面木梁彎曲變形控制方程的基礎(chǔ)上,利用Laplace變換,給出了突加均布載荷作用下FRP布加固簡(jiǎn)支黏彈性木梁彎曲的解析解.根據(jù)相關(guān)試驗(yàn)數(shù)據(jù),確定了花旗松(Douglas-fi r,DF)木梁標(biāo)準(zhǔn)線性固體本構(gòu)的材料參數(shù),分析了芳綸纖維增強(qiáng)塑料(aramid FRP,AFRP)布含量和梁跨高比等對(duì)AFRP布加固簡(jiǎn)支DF木梁彎曲變形的影響.結(jié)果表明: AFRP布加固可有效減小木梁的蠕變撓度;隨著DF木梁蠕變的增加,AFRP布加固DF木梁的中性軸逐漸靠近粘貼AFRP布的側(cè)邊,且隨著DF木梁跨高比或AFRP布含量的提高, AFRP布加固DF木梁的最大壓應(yīng)力和最大拉應(yīng)力減小.

纖維增強(qiáng)聚合物;加固;木梁;黏彈性;解析解

木結(jié)構(gòu)具有自重較輕、節(jié)能環(huán)保、施工方便等優(yōu)點(diǎn),被日本、歐洲和北美等國家和地區(qū)廣泛使用,并且也是國內(nèi)現(xiàn)存寺廟、塔樓等的主要結(jié)構(gòu)形式.作為纖維性材料,木材具有明顯的流變特征,使得木結(jié)構(gòu)在使用中會(huì)產(chǎn)生明顯的蠕變行為.同時(shí),由于木材彈性模量和強(qiáng)度較低,且易腐蝕等,因此木結(jié)構(gòu)及其構(gòu)件需要定期地修復(fù)和加固[1-4].

已有研究表明,采用纖維增強(qiáng)聚合物(fiber reinforced polymer,FRP),如玻璃纖維(glass FRP,GFRP)、碳纖維(carbon FRP,CFRP)和芳綸纖維(aramid FRP,AFRP)等可有效修復(fù)和提高木構(gòu)件的承載力,減小變形,并彌補(bǔ)木構(gòu)件力學(xué)性能老化等缺陷[1].在木梁加固方面,已有學(xué)者對(duì)FRP材料提高木梁的抗彎[1-2,5-9]和抗剪性能[1,10-14]進(jìn)行了廣泛的實(shí)驗(yàn)研究、理論分析和有限元數(shù)值模擬.針對(duì)木梁的蠕變力學(xué)行為,Plevris等[15-16]研究了FRP加固木梁和鋼筋混凝土梁的蠕變行為,而Davids等[17]研究了FRP加固層合木梁的蠕變變形,給出了木材的黏彈性本構(gòu)關(guān)系,并提出了數(shù)值計(jì)算方法.Yahyaei-Moayyed等[18-19]和陸偉東等[20]采用黏性材料的Findley冪律模型研究了木材的蠕變性能,并研究了FRP對(duì)木梁的加固效果.Lu等[21]研究了不同試驗(yàn)環(huán)境和應(yīng)力水平下FRP加固層合木梁的彎曲蠕變,而Pulngern等[22]采用Find ley冪律模型,利用Abaqus有限元軟件研究了木-PVC組合構(gòu)件的短期性能和長(zhǎng)期蠕變行為.可見,已有的研究成果主要是基于Find ley冪律模型,通過試驗(yàn)或有限元數(shù)值仿真研究木梁的黏彈性行為,分析FRP的加固效果.

黏彈性材料的標(biāo)準(zhǔn)線性固體本構(gòu)模型已得到了廣泛應(yīng)用.為此,本工作將木梁視為服從標(biāo)準(zhǔn)線性固體本構(gòu)的黏彈性體,研究底部黏貼FRP布加固木梁彎曲的蠕變行為.基于木梁與FRP布緊密黏貼以及Euler-Bernoulli梁的彎曲變形假定,建立了FRP布加固黏彈性矩形木梁彎曲變形的控制方程,通過Laplace變換及其逆變換,給出了突加均布載荷作用下FRP布加固簡(jiǎn)支黏彈性木梁彎曲的解析解.在此基礎(chǔ)上,利用已有的試驗(yàn)數(shù)據(jù)確定了南美黃松(southern yellow pine,SYP)和花旗松(Douglas-fi r,DF)木材標(biāo)準(zhǔn)線性固體本構(gòu)的材料參數(shù),并分析了AFRP布含量和梁跨高比等對(duì)AFRP布加固簡(jiǎn)支DF木梁彎曲變形蠕變行為的影響.

1 FRP布加固黏彈性矩形木梁彎曲的控制方程

如圖1所示,設(shè)長(zhǎng)為L(zhǎng)、寬為b、高為hI的矩形截面木梁下部貼有FRP布,且承受橫向載荷Q(x,t)的作用,木梁的松弛模量為Y(t),橫截面面積和形心主慣軸的慣性矩分別為AI和II,且形心到木梁底部的距離為rI;FRP布的彈性模量為EII,寬和厚分別b和hII(hII

以木梁軸線為Ox軸,建立直角坐標(biāo)系Oxyz,并記木梁軸線上點(diǎn)的軸向位移為u(x,t),橫向撓度為w(x,t),則根據(jù)變形假定,木梁任一點(diǎn)(x,y,z)的軸向位移、橫向位移和軸向應(yīng)變可分別表示為

圖1 FRP布加固木梁的幾何尺寸F ig.1 Geometric parameters of timber beamstrengthened by FRP sheet

FRP布的軸向位移和軸向應(yīng)變可分別表為

采用黏彈性材料的積分松弛型本構(gòu)方程[23],則木梁和FRP布的軸向應(yīng)力分別為

FRP布橫截面上的軸力FNII為

將FRP布等效為薄膜,記木梁橫截面上的剪力為FSI,木梁與FRP布之間的軸向和橫向作用力分別為Ta和Pa,則由圖2單元體d x平衡可得

圖2 FRP布加固木梁?jiǎn)卧wd xF ig.2 Element d x of timber beamstrengthened by FRP sheet

由式(11)中第一個(gè)方程可得

將其代入式(10)中的第一個(gè)方程得

由式(10)中第二個(gè)方程和式(11)中第一個(gè)方程可得

假定FRP布加固木梁無凈軸力,即FN=FNI+FNII=0,則有

將式(9)代入式(6)中的第三個(gè)方程,可得

方程式(15)和(16)構(gòu)成FRP布加固黏彈性木梁彎曲的基本控制方程.由上述推導(dǎo)可見,對(duì)于其他截面形式的FRP加固木梁,只要將截面幾何參數(shù)進(jìn)行相應(yīng)的替換,控制方程式(15)和(16)仍成立.

2 突加均布載荷作用下FRP布加固簡(jiǎn)支黏彈性木梁的彎曲解析解

假定初始時(shí)FRP布加固黏彈性木梁未變形,則FRP布加固簡(jiǎn)支黏彈性木梁彎曲的初邊值問題為

通過Laplace逆變換,由式(26)和(28)可得FRP布加固簡(jiǎn)支黏彈性木梁彎曲在時(shí)間域的u(x,t)和w(x,t)響應(yīng).然而,對(duì)于任意給定的載荷Q(x,t)和木梁松弛模量Y(t),通常情況下很難得到式(26)和(28)的Laplace逆變換解析解.這里假定木梁為服從標(biāo)準(zhǔn)線性固體本構(gòu)的黏彈性體[23],即松弛模量Y(t)取為

服從標(biāo)準(zhǔn)線性固體本構(gòu)的黏彈性體可視為如圖3所示的黏彈性三參量本構(gòu)模型,其中E1和E2分別為相應(yīng)彈性元件的彈性模型,而η2為黏性元件的黏滯系數(shù),且

式(29)的Laplace變換為

圖3 黏彈性三參量本構(gòu)模型Fig.3 Three-element creepmodel of viscoelasticity

對(duì)均布突加載荷Q(x,t)=Q0H(t),其中Q0為均布載荷的值,H(t)為Heaviside函數(shù). Q(x,t)的Laplace變換為

此時(shí),式(27)化為

而式(22)和(28)化為

對(duì)式(34)進(jìn)行Laplace逆變換,可得

式中,

式中,T為單位時(shí)間,由于本工作黏滯系數(shù)單位取GPa/h,故取T=1 h.FRP布加固木梁軸線的無量綱撓度和水平位移可分別表示為

至此,得到了FRP布加固簡(jiǎn)支黏彈性木梁彎曲變形的解析解式(36)或(40).

進(jìn)一步的分析表明:對(duì)于FRP布加固簡(jiǎn)支黏彈性木梁,只需將式(40)中的因子(ξ4-2ξ3+ ξ)和(2ξ3-3ξ2)分別替換為(ξ4-2ξ3+ξ2)和(2ξ3-3ξ2+ξ),便可得到FRP布加固簡(jiǎn)支黏彈性木梁的彎曲解析解.

3 參數(shù)分析

3.1 木材的蠕變模型

Yahyaei-Moayyed等[19]研究了南美黃松(SYP)以及花旗松(DF)的蠕變行為.試件尺寸為38 mm×38 mm×500 mm.試驗(yàn)采用四點(diǎn)彎曲的加載方式,其中支座間距為432 mm,兩突加集中載荷垂直作用于梁上,且距支座165 mm,間距為102 mm.假定溫度和濕度保持不變,記錄梁底應(yīng)力、應(yīng)變與時(shí)間的相關(guān)數(shù)據(jù).根據(jù)Find ley冪律模型:

給出了材料參數(shù)m,k和n(見表1),然而這些參數(shù)的物理意義欠明確.本工作基于該試驗(yàn)結(jié)果,采用黏彈性體的標(biāo)準(zhǔn)線性固體模型刻畫木材的黏性性質(zhì).黏彈性體標(biāo)準(zhǔn)線性固體本構(gòu)的蠕變形式為

圖4 木梁跨中最外層纖維的應(yīng)變-時(shí)間響應(yīng)Fig.4 Responses of themid-span strains of the outer most fibre vs.time

根據(jù)相關(guān)試驗(yàn)結(jié)果[19],利用最小二乘法可分別擬合SYP和DF材料標(biāo)準(zhǔn)線性固體模型的本構(gòu)參數(shù)E1,E2和η2.圖4分別給出了SYP和DF的試驗(yàn)數(shù)據(jù)及Find ley冪律模型和標(biāo)準(zhǔn)線性固體模型的擬合曲線,其中實(shí)線為標(biāo)準(zhǔn)線性固體本構(gòu)模型式(39)的擬合結(jié)果,而虛線為Find ley冪律模型擬合結(jié)果.可見,兩種蠕變模型均能很好地與試驗(yàn)數(shù)據(jù)吻合,并且對(duì)取自同一木料制成的DF試件的蠕變擬合呈現(xiàn)高度一致.表1給出了兩種模型本構(gòu)參數(shù)的擬合結(jié)果.

表1 SYP和DF木材的Find ley冪律模型和標(biāo)準(zhǔn)線性固體模型參數(shù)Tab le 1 Model parameters of Find ley’s power law and standard linear solid model of the SYP and DF woods

3.2 參數(shù)分析

由于現(xiàn)有的取自同一木料的試驗(yàn)數(shù)據(jù)很少,故選取各物理參數(shù)的平均值來分析FRP布加固木梁的蠕變響應(yīng).為重點(diǎn)考察FRP布加固對(duì)簡(jiǎn)支黏彈性木梁力學(xué)性能的影響,取Q?=1×10-3,hI=b=0.1 m,并采用表1中DF材料參數(shù)的平均值,即E1=247.72 GPa,E2= 42.62 GPa,η2=11.60 GPa·h,以及文獻(xiàn)[19]中AFRP布受荷500 h的彈性模量EII= 124GPa進(jìn)行計(jì)算和分析.此時(shí),無量綱材料參數(shù)為

加固木梁AFRP含量定義為H2=hII/h,即AFRP布加固木梁橫截面中AFRP布的面積占加固木梁橫截面總面積的百分?jǐn)?shù),由此可得無量綱幾何參數(shù)H1=1-H2,γ3= (1-H2)/H2,γ4=3γ3.

圖5給出了跨高比L/hI=10時(shí),不同AFRP含量H2下AFRP布加固簡(jiǎn)支DF梁跨中無量綱撓度W?=W(0.5,t?)隨無量綱時(shí)間t?的響應(yīng).圖6給出了不同無量綱時(shí)刻t?時(shí),未加固DF梁(H2=0)和含量H2=2.91%的AFRP布加固DF梁無量綱撓度W=W(ξ,t?)沿梁軸線的分布,其中虛線和實(shí)線分別為未加固DF梁和AFRP布加固DF梁的撓度分布.可見,對(duì)給定跨高比L/hI,無量綱撓度W?隨無量綱時(shí)間t?的增加而增大,并趨于穩(wěn)定值,并且無量綱撓度W?隨AFRP布含量的增加而減小.當(dāng)t?=0時(shí),H2分別為0.99%,1.96%和2.91%的AFRP布加固木梁的彈性撓度W?分別比未加固木梁(即H2=0)降低1.46%,2.92%和4.22%;而當(dāng)無量綱撓度W?的蠕變達(dá)到穩(wěn)定值,即t?→∞時(shí),AFRP布加固木梁撓度W?分別比H2=0時(shí)降低9.00%,16.07%,21.78%,即AFRP加固可有效減小木梁的蠕變撓度.同時(shí),從解析解(40)可以發(fā)現(xiàn),無量綱撓度W?的減小量不依賴于無量綱載荷Q?、跨高比L/hI和梁寬b.

圖5 不同AFRP布含量H2時(shí),AFRP布加固DF梁無量綱跨中撓度W?隨無量綱時(shí)間t?的響應(yīng)Fig.5 Responses of non-dimensionalmid-span deflections W?vs.non-dimensional time t?of AFRP-reinforced DF beamfor volume fractions H2of AFRP

圖6未加固DF梁與當(dāng)H2=2.91%時(shí),AFRP布加固DF梁無量綱撓度W(ξ,t?)的分布Fig.6 Distributions of non-dimensional deflections W(ξ,t?)of unreinforced and AFRP-rein forced DF beamwhen H2=2.91%

圖7 給出了不同無量綱時(shí)刻t?時(shí),未加固DF梁和H2=2.91%的AFRP布加固DF梁端部ξ=1處橫截面無量綱軸向位移U(ζ,t?)=ut/L沿DF梁無量綱高度坐標(biāo)ζ=y/h的分布.可見,隨著無量綱時(shí)間t?的增大,未加固DF梁和AFRP布加固DF梁的端部軸向位移(橫截面轉(zhuǎn)角)均逐漸增大,但對(duì)未加固DF梁,其中性軸位置不變,而對(duì)于AFRP布加固DF梁,其中性軸逐漸靠近粘貼AFRP布的側(cè)邊.

圖7不同無量綱時(shí)刻t?時(shí),未加固DF梁和AFRP布加固DF梁端ξ=x處無量綱軸向位移U(ζ,t?)沿?zé)o量綱高度ζ的分布Fig.7 Distributions of the non-dimensional axial displacements U(ζ,t?)at endξ=x of the unreinforced and AFRP-reinforced DF beamalong the non-dimensional high coordinateζfor diff erent non-dimensional instant time t?

圖8 和9分別給出了不同跨高比L/hI和AFRP含量H2下,AFRP布加固DF梁跨中無量綱最大壓應(yīng)力Σc=σI/E1|ζ=-rI,ξ=0.5和無量綱最大拉應(yīng)力Σt=σI/E1|ζ=rI,ξ=0.5隨無量綱時(shí)間t?的響應(yīng).可見,對(duì)給定的跨高比L/hI,隨著無量綱時(shí)間t?的增加,AFRP布加固DF梁的無量綱最大壓應(yīng)力Σc和無量綱最大拉應(yīng)力Σt的數(shù)值均逐漸減小,并趨于穩(wěn)定值,拉應(yīng)力Σt的變化幅度大于壓應(yīng)力Σc.同時(shí),隨著AFRP布加固DF梁跨高比L/hI或AFRP布含量H2的提高,DF梁的最大壓應(yīng)力Σc和最大拉應(yīng)力Σt均減小.這是因?yàn)殡S著梁跨高比L/hI或AFRP布含量H2的提高,AFRP布承擔(dān)的載荷增加,從而導(dǎo)致DF梁應(yīng)力的減小.

圖10分別給出了不同跨高比L/hI和AFRP布含量H2下,AFRP布加固DF梁跨中AFRP布無量綱拉應(yīng)力Σf=σII/E1|ζ=rI,ξ=0.5隨無量綱時(shí)間t?的響應(yīng).可見,對(duì)給定的跨高比L/hI,隨著無量綱時(shí)間t?的增加,AFRP布的無量綱拉應(yīng)力Σf增大,并趨于穩(wěn)定值,并且隨著AFRP布含量H2的提高和跨高比L/hI的減小,無量綱拉應(yīng)力Σf增大.

4 結(jié)論

本工作考慮了木梁的黏彈性性質(zhì),研究了FRP布加固木梁的線性彎曲行為.首先,在木梁材料服從標(biāo)準(zhǔn)線性固體本構(gòu)和木梁與FRP布緊密粘貼的假定下,建立了FRP布加固黏彈性矩形木梁彎曲變形的控制方程,通過Laplace變換及其逆變換,給出了突加均布載荷作用下FRP布加固簡(jiǎn)支黏彈性木梁彎曲的解析解.然后,在此基礎(chǔ)上,基于相關(guān)的試驗(yàn)數(shù)據(jù),確定了南美黃松(SYP)和花旗松(DF)木材標(biāo)準(zhǔn)線性固體本構(gòu)的材料參數(shù),并分析了AFRP布含量和梁跨高比等對(duì)AFRP布加固DF梁彎曲變形蠕變行為的影響.最后,得到如下結(jié)論.

(1)木材黏性對(duì)木梁的彎曲力學(xué)性能有顯著的影響,AFRP布加固木梁不僅可減小木梁初始彈性撓度,而且可有效減小其蠕變撓度;

(2)隨著時(shí)間增加,未加固或AFRP布加固DF梁端部轉(zhuǎn)角均逐漸增大,但對(duì)于AFRP布加固DF梁,其中性軸逐漸靠近粘貼AFRP布的側(cè)邊.

(3)隨著時(shí)間增加,AFRP布加固DF梁的最大壓應(yīng)力和最大拉應(yīng)力均逐漸減小,并趨于穩(wěn)定值,且隨著DF梁跨高比或AFRP含量的提高,AFRP布加固DF梁的最大壓應(yīng)力和最大拉應(yīng)力減小.

(4)隨著AFRP含量H2的提高和跨高比L/hI的減小,AFRP布拉應(yīng)力Σf增大.

圖8 不同跨高比L/hI和AFRP布含量H2下,AFRP布加固DF梁無量綱最大壓應(yīng)力Σc隨無量綱時(shí)間t?的響應(yīng)Fig.8 Responses of non-dimensionalmaximumcompressive stressΣcof AFRP-reinforced DF beamvs.non-dimensional time t?for diff erent span-depth ratio L/hIand volume fraction H2of AFRP

圖9 不同跨高比L/hI和AFRP含量H2下,AFRP布加固DF梁無量綱最大拉應(yīng)力Σt隨無量綱時(shí)間t?的響應(yīng)Fig.9 Responses of non-dimensionalmaximumtensional stressΣtof AFRP-reinforced DF beamvs.non-dimensional time t?for diff erent span-depth ratio L/hIand volume fraction H2of AFRP

圖10 不同跨高比L/hI和AFRP含量H2下,AFRP布加固DF梁中AFRP布無量綱拉應(yīng)力Σf隨無量綱時(shí)間t?的響應(yīng)Fig.10 Responses of non-dimensional tensional stressΣfof AFRP sheet in AFRP-rein forced DF beamvs.non-dimensional time t?for diff erent span-depth ratio L/hIand volume fraction H2of AFRP

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Analy tical solu tion of bend ing of v iscoelastic timber beamrein forced with fi ber rein forced po lymer sheet

OUYANG Yu,JIANG Yong,ZHOU Lei
(Department of Civil Engineering,Shanghai University,Shanghai 200444,China)

Regarding timber beamas a viscoelastic mediumwith a standard linear solid constitutive relation,and assuming bonding tightly between the fiber reinforced polymer (FRP)sheet and timber beam,the creepbehavior of linear bending of the timber beamreinforced with FRP sheet is studied.Based on the estab lished governing equation for bending deformation of the viscoelastic rectangular cross-section timber beamreinforced with FRP sheet,ananalytical solution of bending of the simply-supported FRP-reinforced viscoelastic timber beamsub ject to stepuniformload is presented by using Laplace transform.The material parameters of the standard linear solid constitutive relation for Douglas-fi r(DF) timber are determined with existing experimental data.The influences of volume fraction of aramid FRP(AFRP)sheet and span-depth ratio of beamon the bending behavior of the simply-supported DF timber beamreinforced with AFRP sheet is analyzed numerically. It is shown that creepdeflections of the DF timber beamcan be eff ectively decreased by AFRP sheet reinforcement.W ith development of creepof the DF timber beam,the neutral axis of the DF timber beamreinforced with AFRP sheet moves to the edge of the timberbeamof bonding AFRP sheet.Furthermore,with increase of the span-depth ratio of DF timber beamand volume fraction of AFRP sheet,maximumcompressive and tensional stresses of the AFRP-reinforced DF timber beamdecrease.

fiber reinforced polymer(FRP);reinforcement;timber beam;viscoelasticity; analytical solution

TU 323.3;O 344.6

A

1007-2861(2017)04-0609-14

DO I:10.12066/j.issn.1007-2861.1664

2015-09-07

國家高技術(shù)研究發(fā)展計(jì)劃(863計(jì)劃)資助項(xiàng)目(2009AA032303-2)

歐陽煜(1968—),男,副教授,博士,研究方向?yàn)榻M合結(jié)構(gòu)、結(jié)構(gòu)加固與修復(fù)研究.

E-mail:Oyy w ly@sina.com