第4章内力计算

第4章内力计算4.1风荷载作用下框架内力计算由D值法计算结构在风荷载作用下的内力，计算过程及结果见表4-1。表4-1风荷载作用下柱端弯矩表(kNm)层次边柱D/EDV.1y.M上M下50.2338.450.940.44653.272.6440.23318.481.270.457.105.8130.23327.971.490.59.789.7820.23336.571.860.512.7812.7810.24244.23.210.5521.9026.77层次中柱D/EDV.1y.M上M下50.2678.451.410.453.723.0540.26718.481.490.57.407.4030.26727.971.490.511.2011.2020.26736.571.650.514.6514.6510.25844.212.470.5523.3528.54梁端风荷载弯矩、剪力、轴力 见表4-2，其中入=Kr/ (Kr+KL)。4.2水平地震作用下框架内力计算以12轴横向框架为例进行计算,在水平地震作用下的框架住弯矩计算采用D值法, 其计算过程见表4-3。表4-2梁端风荷载弯矩(kNm)、剪力(kN)、轴力(kN)层次边柱处中柱处剪力轴力EMcEMc1-入入EMc右EMc左ABBC边柱中柱续表4-2层次边柱处中柱处剪力轴力EMcEMc1-入入EMc右EMc左ABBC边柱中柱53.273.720.310.692.571.150.822.140.821.3249.7410.450.310.697.213.242.406.013.224.93312.5918.600.310.6912.835.773.4010.696.6212.22222.5625.850.310.6917.848.015.6614.8712.2821.43134.6838.000.310.6926.2211.788.6021.8520.8834.68风荷载作用下框架的弯矩、剪力、轴力图见图4-1,4-2。图4-1风荷载作用下的框架弯矩（单位KN.m）图4-2风荷载作用下的剪力、轴力（单位KNm）表4-3水平地震作用下框架柱弯矩计算柱层次层剪力总刚度各柱刚度D/EDVikyhM下M上边柱5748.611102920168700.015311.452.930.44715.3419.0141366.511102920168700.015320.912.930.4528.2334.5031847.601102920168700.015328.272.930.542.4142.41续表4-322191.891102920168700.015333.542.930.550.3150.3112470.0740992062300.015237.554.460.5593.9776.88中柱5748.611102920192800.017513.104.240.4517.6921.6241366.511102920192800.017523.914.240.535.8735.8731847.601102920192800.017532.334.240.548.5048.5022191.891102920192800.017538.364.240.557.5457.5412470.0740992066700.016340.266.450.55100.7582.43水平地震作用下框架剪力、轴力计算见表4-4。表4-4水平地震作用下框架剪力、轴力层次边柱中柱剪力轴力Me梁端2MeMrMlabbe边柱中柱519.01-19.010.6921.6214.926.704.7612.434.767.67449.84-49.840.6953.5636.9616.6012.3030.817.0626.17370.64-70.640.6984.3758.2226.1517.9248.5234.9856.77292.72-92.720.69106.0473.1732.8723.2660.9858.2494.491127.19-127.190.69139.9796.5843.3931.5980.4889.83143.38地震作用下框架的弯矩、剪力、轴力图见图4-3, 4-4。B9.93图4-4水平地震作用下的剪力、轴力图（单位kN）4.3恒载作用下框架内力计算4. 3. 1计算方法选用本工程荷载作用下的内力计算采用分层法4.3.2用弯矩分配法计算梁柱端弯矩注意：除底层外柱的线刚度需要乘以修正系数0.9，并且除底层柱外其它各层的弯 矩传递系数取1/3。修正后的梁柱线刚度见表4-5。表4-5梁柱线刚度表（10-E .m3）层次梁柱iAB(iCD)iBCiC2-56.9520.837.1116.9520.834.22中间层A点分配系数4 x 7.114 x 7.11 + 4 x 7.11 + 4 x 6.95二 0.3364i卩 二 上 上4i + 4i + 4i上 下 AB卩二卩二 0.336下 上卩 二 1 一卩一卩 二0.328AB上 下中间节点B节点分配系数4i上一4i + 4i + 4i + 2i上 下 AB BC4 x 7.114 x 7.11 + 4 x 7.11 + 4 x 6.95 + 2 x 20.83二 0.225卩二卩二 0.225下 上4i卩 二ABAB 4i + 4i + 4i + 2i上 下 AB BC4 x 6.954 x 7.11 + 4 x 7.11 + 4 x 6.95 + 2 x 20.83二 0.22|L1 1 |L1 |L1 |L1 0.33BC上 下 AB底层A点分配系数4i上4i + 4i + 4i上下AB4 x 7.114 x 7.11 + 4 x 4.22 + 4 x 6.950.3894i下4i + 4i + 4i上下AB4 x 4.224 x 7.11 + 4 x 4.22 + 4 x 6.950.231卩1 一卩一卩0.380AB上 下底层B点分配系数4i上4i + 4i + 4i + 2i上 下 AB BC4 x 7.114 x 7.11 + 4 x 4.22 + 4 x 6.95 + 2 x 20.830.2484i下4i + 4i + 4i + 2i上 下 AB BC4 x 4.224 x 7.11 + 4 x 4.22 + 4 x 6.95 + 2 x 20.830.1474iAB4i + 4i + 4i + 2i上 下 AB BC4 x 6.954 x 7.11 + 4 x 4.22 + 4 x 6.95 + 2 x 20.830.242|L1 1 一 |L1 一 |L1 一 |L1 0.363BC上 下 AB顶层A点分配系数4iU 下 -下 4i + 4i下AB4 x 7.114 x 7.11 + 4 x 6.950.5060.291卩1 一 卩0.494AB下顶层B点分配系数4i卩 下4i + 4i+ 2i下 AB BC4 x 7.114 x 7.11 + 4 x 6.95 + 2 x 20.834i卩 二ABAB 4i + 4i+ 2i下ABBC4 x 6.954 x 7.11 + 4 x 6.95 + 2 x 20.83二 0.284|L1= 1 |L1 |L1= 0.425BC下 AB固端弯矩及弯矩分配系数见表4-6。表4-6分层法分配系数及恒载作用下的固端弯矩计算结果节点单兀ABEA上柱A下柱AB端BA端B上柱B下柱BE端E端分配系数顶层0.5060.4940.2840.2910.425中间层0.3360.3360.3280.220.2250.2250.33底层0.3890.2310.380.2420.2480.1470.363-固端弯矩顶层-50.3750.37-6.72-3.36中间层-70.4770.47-5.30-2.65底层-70.4770.47-5.30-2.65分配法计算过程见图4-5图4-7。4. 3. 3跨中弯矩计算根据求得的支座处弯矩，按平衡条件计算，框架梁在实际荷载作用下按简支梁计算 的跨中弯矩M0,并查静力手册，在实际荷载分布下，框架的跨中弯矩计算按M=M-（M 左+M 右） /2。结果见表4-7。表4-7实际荷载分布下分层法各单元跨中弯矩计算表（kN m）位置按简支计算跨中弯矩叠加后的跨中弯矩左端右端跨中AB、CD 跨顶层75.5629.6748.2936.58中间层105.7153.1366.2346.03底层105.7149.7366.2247.74BC跨顶层10.0931.4031.40-21.31中间层7.9531.0831.08-23.13底层7.9534.4834.48-26.53-50.370.550.37、1 d a-6,721-3；3624,88 -0.5才11 i n- n-2S；84 -1 23;840 s-15,933,93T 1,97-10.5-0.84 -0,8-4-0.28 T -0.560,14-29,67斗 8,29-31,斗21,320,494-70,47贯11-8,44E.77-0,15/0.5旷D.5、丿 0,50,570,4711.56-16.881.3S-0.3.一阴-2,65-25/32 厶532口洱& -二0.46-53.13-31,0823,1326.57-TS30.22m0.2250.052,843,680；3360毕3168 0,3360.2250.3:2.S40,0526,57173ABQ Q-inn8,86I1Z3-5.86A1/3-17.57-0,31_-17,26-17,E6 -0.31 -17.57 l-Z-3-5,36图4-6中间层弯矩分配法计算过程（单位kN m）-70.47-0.EE尸0.5、0,5:0,526.78-9.513.610,570,4713.39-19.011,81-0.44-49173-5,3-E.65-28.52 -10.66 -1t 0.66-34,482G.5310.40T1ZS.31.200.093,7(?27.410890.381G.282.200,050.2311S.531/E0陆斗2EE0.2480,1470.363E6.64/tx173-19.93-0.45-19.48吕-11.55-0.E7-11.821/2-5：91图4-7底层弯矩分配法计算过程（单位kN m）4.3.4由各层单元计算整个结构在恒载作用下的弯矩将分层法求得的各层弯矩图叠加，可得整个框架在竖向荷载的弯矩图。很显然 叠加后各框架节点弯矩并不能达到平衡，这是由于分层计算的误差造成的。为了提高精 度，可将不平衡弯矩分配进行一次修正，叠加后的框架梁跨中弯矩计算表见表4-8。表4-8叠加后的框架梁跨中弯矩计算位置按简支计算跨中弯矩叠加后跨中弯矩左端右端跨中AB跨（CD 跨）575.5634.0549.9533.564105.7150.5762.7649.053105.7158.9568.8141.832105.7159.4568.9841.501105.7153.1067.6445.34BC跨510.0932.1032.10-22.0147.9536.2836.28-28.3337.9534.9434.94-26.9927.9535.2035.20-27.2517.9536.6136.61-28.664. 3. 5梁端剪力及梁端控制界面的弯矩 梁端剪力计算用平衡条件可求出的梁端剪力及梁端柱边剪力，计算结果见表4-9。 梁端柱边剪力计算取柱轴心至柱边这一端梁为隔离体，由平衡条件可求出的梁端柱边的剪力值，对AB跨梁取图 为隔离体，由竖向力平衡及几何条件，可得V = V - 0.2g= V - 0.2(g + 0.5g )AB ABAB AB12V = V - 0.2g= V - 0.2(g + 0.5g )BA BAAB BA12V = V = V - 0.2g = V - 0.2(g + 0.5g )BC CB BCAB BC12 梁端柱边弯矩计算为内力组合做准备，需将梁端弯矩换算为梁端柱边弯矩，对本工程按下式计算M丄M-0.2VABABABM =M-0.2V BABABAM=M-0.2VBCBCBC表4-9梁端弯矩(kN m)、剪力(kN)AB跨(CD跨)层次glg2MabMbaVabVbaVabVba53.2617.4734.0549.9523.5641.8821.1637.34416.3612.6450.5762.7645.5568.2341.0163.69316.3612.6458.9568.8148.6667.8044.1263.26216.3612.6459.4568.9848.8467.7444.3063.20116.3612.6453.1067.6446.4968.6741.9564.13BC跨层次g1g2VbcVcbVbcVcb53.2610.7519.3919.3917.6617.6643.267.7815.1215.1213.6913.6933.267.7815.1215.1213.6913.6923.267.7815.1215.1213.6913.6913.267.7815.1215.1213.6913.69梁端柱边剪力、弯矩计算结果见表4-10。表4-10梁端柱边剪力(kN)、弯矩(kN m)计算层次梁端弯矩梁端柱边剪力梁端柱边弯矩MabMbaMbcVabVbaVbcM /abM baM bc534.0549.9532.1021.1637.3417.6629.8242.4828.57450.5762.7636.2841.0163.6913.6942.3750.0233.26358.9568.8134.9444.1263.2613.6950.7556.1631.92259.4568.9835.2044.3063.2013.6950.5956.3432.18153.1067.6436.6141.9564.1313.6944.7154.8133.594. 3. 6柱轴力计算柱轴力可通过对梁端剪力、纵向梁传来剪力和柱自重叠加得到。柱轴力的计算过程 见表4-11。表4-11柱轴力计算(kN)Vab竖向力F柱自重轴力VbaVbc竖向力柱自重轴力23.5648.1512顶71.7141.8819.3944.2712顶105.54底83.71底117.5445.5569.9012顶115.4568.2315.1284.7512顶168.10底127.45底180.1048.6669.9012顶118.5667.8015.1284.7512顶167.67底130.56底179.6748.8469.9012顶118.7467.7415.1284.7512顶167.61底130.74底179.6146.4969.9018.20顶116.3968.6715.1284.7518.4顶168.54底134.59底186.944.3.7弯矩调幅本工程考虑梁端弯矩调幅，调幅系数0 = 0.85，恒载作用下的梁剪力、柱轴力见图4-8，调幅后的梁端柱边弯矩及跨中弯矩见图4-9。调幅后的梁端柱边弯矩及跨中弯矩见表4-12。表4-12调幅后的梁端、跨中弯矩层次AB跨（CD跨）BC跨MabMmMbaMbcMm525.3544.8336.1124.28-14.19436.0166.4442.5228.27-20.32343.1460.2747.7427.13-19.18243.0060.2647.8927.35-19.40138.0063.4146.5928.55-20.60图4-8恒载作用下的剪力、轴力图（单位：kN）图4-9恒载作用下经调幅的柱边弯矩图（单位：kNm）4.4活载作用下的内力计算活载同时作用于所有框架梁上，按满荷载分布考虑，不考虑活载的不利布置并采用 分层法，跨中弯矩乘以1.1-1. 2的放大系数，以考虑其不利布置可能产生的影响。4.4.1用弯矩分配法计算梁端弯矩顶层：1M =一x 7.8 x 5.42 =18.95kN - mAB 12M =- M= 18.95kN - mBAAB1M =-x 4.8 x 1.22 =-2.30kN - mBE 31M =- x 4.8 x 1.22 =-1.15kN - mEB6其它层：1M =一x 7.8 x 5.42 =18.95kN - mAB 12M =- M= 18.95kN - mBAAB1M =x 6.0x 1.22 =2.88kN - mBE 31M = - X 6.0 x 1.22 =_1.44kN - mEB 64. 4. 2内力计算活载作用下的梁端弯矩、剪力、柱轴力计算过程与恒载作用下的计算过程完全一样。 计算过程不再累述。计算结果见表4-13。表4-13梁端柱边剪力计算(kN)ab跨梁端柱边剪力计算层次g1g2MabMbaVabVbaVJVba57.8012.722.0815.2322.8013.6721.2447.8016.2219.0116.5421.5814.9820.0237.8015.7418.7516.3621.6214.8020.0627.8015.8718.8017.4921.6015.9320.0417.8014.1718.4815.7821.8614.2220.30be跨梁端柱边剪力计算层次g1g2MbeVbeVbeVeb54.8013.175.764.84.846.007.977.26.06.036.008.367.26.06.026.008.297.26.06.016.009.657.26.06.0活荷载作用下的梁端柱边弯矩计算见表4-14。表4-14活荷载作用下的梁端柱边弯矩计算层次梁端弯矩梁端柱边剪力梁端柱边弯矩MabMbaMbeVabVbaVbeMabMbaMbe512.722.0813.1713.6721.244.89.9717.8312.21416.2219.017.9714.9820.026.013.2215.016.77315.7418.758.3614.8020.066.012.7814.747.16215.8718.808.2915.9320.046.012.6814.797.09114.1718.489.6514.2220.306.011.3314.428.454. 4. 3弯矩调幅考虑梁端弯矩调幅，调幅系数0二0.85 ,调幅后的梁端柱边弯矩及跨中弯矩见图4T2。表4-15调幅后的梁端、跨中弯矩(kN m)层次AB跨(CD跨)BC跨MabMMbaMbcM58.47116.61615.15710. 379-6. 919411.2416. 43212. 7555. 755-1.435310.86316. 73512. 5276. 086-1. 766210.78116. 75312. 5736. 027-1. 70719. 62717.48812. 2577. 183-2. 863风荷载作用下的柱边弯矩、剪力计算见表4-16。表4-16风荷载作用下的柱边弯矩、剪力计算层次梁端弯矩梁端剪力梁端柱边弯矩MabMbaMbcVabVbaVbcMabMbaMJ53.271.152.570.820.822.143.110.992.1449.743.247.212.402.406.019.262.766.01312.595.7712.833.403.4010.6911.915.0910.69222.568.0117.845.665.6614.8721.436.8814.87134.6811.7826.228.608.6021.8532.9610.0621.85风荷载作用下的柱边弯矩见图4-10。地震作用下的柱边弯矩计算见表4-17。水平地震作用下的柱边弯矩见图4-11。表4-17地震作用下梁端柱边弯矩(kN m)、剪力(kN)计算层次梁端弯矩梁端剪力梁端柱边弯矩MabMbaMbcVabVbaVbcMIMbaMbc519.016.7014.924.764.7612.4318.065.7512.43449.8416.6036.9612.3012.3030.8047.3814.1430.80370.6426.1558.2217.9217.9248.5267.0622.5748.52292.7232.8773.1723.2623.2660.9888.0728.2260.971127.1943.3996.5831.5931.5980.48120.8737.0780.48图4-10风荷载作用下柱边弯矩图（kNm）5.74B&7.0562341H.D2Z.56tL-/51620.210/50J74/ee.-DhB120.672图4-11水平地震作用下的柱边弯矩图（kNm）图4-12活载作用下经调幅后的柱边弯矩图（kNm）