梁板式楼梯计算书

梁板式楼梯:

一,基本资料:

工程名称:工程一
楼梯类型:板式 A 型 ( ╱ ) 支座条件:两端弹性

梯段水平投影长度Lo1 = 4200mm 梯段净跨Lo = Lo1 = 4200mm
梯板净宽度B = 2000mm
梯板厚度h1 = 160mm

梯段高度Hv = 2250mm 楼梯级数n = 15
下支座宽度d1 = 250mm 上支座宽度d2 = 250mm

线性恒荷Pk = 0.00kN/m
均布活荷qk = 3.50kN/m? ψc = 0.7 ψq = 0.4
面层厚度c1 = 20mm 顶棚厚度c2 = 20mm
面层,顶棚容重 γc =20.00kN/m? 楼梯自重容重γb = 25.00kN/m?

混凝土强度等级:C30
fc = 14.33 ft = 1.43 ftk = 2.01N/mm? Ec = 29791N/mm?
钢筋强度设计值fy = 360N/mm?Es = 200000N/mm?
纵筋合力点至近边距离as = 20mm

二,计算结果:

1,楼梯几何参数:
踏步高度t = Hv / n = 2250/15 = 150.0mm
踏步宽度w = Lo1 / (n - 1) = 4200/(15-1) = 300.0mm
梯段斜板的倾角α = ArcTan(t / w) = ArcTan(150.0/300.0) = 26.6°
梯段斜板的长度Lx = Lo1 / Cosα = 4200/Cos26.6°= 4696mm
梯板板厚的垂直高度h1' = h1 / Cosα = 160/Cos26.6°= 179mm
梯板平均厚度T = (t + 2 * h1') / 2 = (150+2*179)/2 = 254mm
梯板有效高度h1o = h1 - as = 160-20 = 140mm

2,荷载设计值:
(1),均布恒荷标准值:
①,梯板自重:
gk1 = γb * T = 25.00*0.254 = 6.35kN/m?
②,Lo1段梯板面层自重:
gk2 = γc * c1 * [n * t + (n - 1) * w] / Lo
= 20.00*0.020*[15*0.150+(15-1)*0.300]/4.200 = 0.61kN/m?
③,梯板顶棚自重: gk3' = γc * c2 = 20.00*0.020 = 0.40
gk3 = gk3' * Lx / Lo = 0.40*4.696/4.200 = 0.45kN/m?

gk = gk1 + gk2 + gk3 = 7.41kN/m?

(2),均布荷载设计值:
由活荷控制的荷载设计值p(L) = 1.2 * gk + 1.4 * qk = 13.79kN/m?
由恒荷控制的荷载设计值p(D) = 1.35 * gk + 1.4 * ψc * qk = 13.43kN/m?
最不利的荷载设计值p = Max{p(L), p(D)} = 13.79kN/m?

3,斜截面受剪承载力计算:
Vmax ≤ 0.7 * ft * b * ho
Vmax = 0.5 * p * Lo * Cosα = 0.5*13.79*4200*Cos26.6°
= 28960*0.8944 = 25902N
0.7 * ft * b * ho = 0.7*1.43*1000*140 = 140423N
≥ Vmax = 25902N,满足要求.

4,正截面受弯承载力计算:
计算跨度L = Min{Lo + (d1 + d2) / 2, 1.05 * Lo}
= Min{4450, 4410} = 4410mm
跨中Mmax = p * L ^ 2 / 10 = 13.79*4.410^2/10 = 26.820kN·M
As = 560mm? ρ = 0.40%
φ10@100 (As = 785) ωs,max = 0.168mm

支座Mmin = p * L ^ 2 / 24 = 13.79*4.410^2/24 = -11.175kN·M
As = 226mm? ρ = 0.16% ρmin = 0.20% As* = 320mm?
φ10@200 (As = 393) ωs,max = 0.037mm

5,跨中挠度验算:
Mk -------- 按荷载效应的标准组合计算的弯矩值
Mq -------- 按荷载效应的准永久组合计算的弯矩值

(1),挠度验算参数:
Mk = 21.22kN·M Mq = 17.13kN·M
(2),在荷载效应的标准组合作用下,受弯构件的短期刚度Bs:
①,裂缝间纵向受拉钢筋应变不均匀系数ψ,按下列公式计算:
ψ = 1.1 - 0.65 * ftk / ρte / σsk (混凝土规范8.1.2-2)
σsk = Mk / 0.87 / ho / As (混凝土规范8.1.3-3)
σsk = 21215223/0.87/140/785 = 222N/mm?
矩形截面,Ate = 0.5 * b * h = 0.5*1000*160 = 80000mm?
ρte = As / Ate (混凝土规范8.1.2-4)
ρte = 785/80000 = 0.98%
ψ = 1.1 - 0.65*2.01/0.00981/222 = 0.501
②,钢筋弹性模量与混凝土模量的比值αE:
αE = Es / Ec = 200000/29791 = 6.71
③,受压翼缘面积与腹板有效面积的比值γf':
矩形截面,γf' = 0
④,纵向受拉钢筋配筋率ρ = As / b / ho = 785/1000/140 = 0.00561
⑤,钢筋混凝土受弯构件的 Bs 按公式(混凝土规范8.2.3-1)计算:
Bs = Es * As * ho ^ 2 / [1.15ψ + 0.2 + 6 * αE * ρ / (1 + 3.5γf')]
Bs = 200000*785*140^2/[1.15*0.501+0.2+
6*6.71*0.00561/(1+3.5*0.000)] = 3070.64kN·M?

(3),考虑荷载长期效应组合对挠度影响增大影响系数θ:
按混凝土规范第 8.2.5 条,当ρ' = 0时,θ = 2.0

(4),受弯构件的长期刚度B,可按下列公式计算:
B = Mk / [Mq * (θ - 1) + Mk] * Bs (混凝土规范8.2.2)
B = 21.22/[17.13*(2-1)+21.22]*3070.64 = 1698.84kN·M?

(5),跨中挠度f = 3.0 / 384 * Qk * L ^ 4 / B
f =0.00781*10.91*4.410^4/1698.84*1000 = 19.0mm
f / Lo = 1/221