响应谱分析

结构分析方法,用于预测结构在承受动态载荷(通常是地震或爆炸等瞬态事件)时的最大响应。

响应谱分析 (RSA) 可估算出结构在这些条件下所承受的最大位移、应力和力。RSA 将特定动态载荷的预定义响应谱与法向模态分析结果相结合。与时间历史分析不同,RSA 不会生成响应的时间演变。

响应谱说明了单自由度 (1-DOF) 系统在特定动态载荷下的最大响应与固有频率的关系。这些谱有助于计算每个结构模态的最大模态响应。各种组合方法,如绝对和 (ABS),平方和的平方根 (SRSS),海军研究实验室 (NRL) 或完全二次项组合 (CQC),合并这些模态最大值来估算结构的峰值响应。

与传统的瞬态分析方法相比,RSA 提供了一种更简单、计算效率更高的近似峰值响应方法。主要的计算要求在于获取足够数量的法向模态,以准确表示输入激励和结果响应的整个频率范围。通常,设计规范会提供响应谱,以便快速计算不同动态激励下的峰值响应。因此,RSA 是一种常用的设计工具,特别是在建筑物的抗震分析中。

在对结构执行 RSA 时,选择适当的模态组合方法对于获得准确的结果至关重要。这种方法决定了软件如何将单个振动模式(模态响应)的原始结果合并为每一自由度的位移、反作用力、内力和其他参数的单一值集。这些综合结果是设计结构的基础,突出了谨慎选择模态组合方法的重要性。下文将探讨 RSA 中常用的几种模态组合方法。

模态组合

绝对和 (ABS)

绝对和模态组合法会计算每个振动模式结果(如位移或内力)的绝对值,并将这些绝对值相加。这种方法会假设所有峰值模态响应同时发生,因此估算结果比较保守。因此,由于其过于保守的性质,它在结构设计应用中并未得到广泛青睐。

采用绝对和法计算总响应峰值的公式为:

总响应峰值 = Σ i R n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabw gacaqGHbGaae4AaiaabccacaqGsbGaaeyzaiaabohacaqGWbGaae4B aiaab6gacaqGZbGaaeyzaiaabccacaqGubGaae4BaiaabshacaqGHb GaaeiBaiabg2da9iabfo6atnaaBaaaleaacaWGPbaabeaakmaaemaa baGaamOuamaaBaaaleaacaWGUbaabeaaaOGaay5bSlaawIa7aaaa@4F94@
R n = A i n ψ i X MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGUbaabeaakiabg2da9iaadgeadaWgaaWcbaGaamyAaiaa d6gaaeqaaOGaeqiYdK3aaSbaaSqaaiaadMgaaeqaaOGaamiwaaaa@3FA9@
其中,

R n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGUbaabeaaaaa@37ED@
代表每个振动模式在单个自由度下的瞬态响应
A MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
特征向量
ψ MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdKhaaa@37C5@
模态参与因子
X MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36D4@
输入响应谱
这个简单明了的公式会将所有模式响应的绝对值相加,而不考虑它们的符号。

平方和的平方根 (SRSS)

SRSS 模态组合方法会计算每种振动模式结果平方和的平方根,从而得出总响应峰值的近似值。这种方法对于具有不同固有频率的结构尤为有效。但是,如果结构的固有频率间隔很近,SRSS 可能无法得出准确的结果,应避免使用。

采用 SRSS 方法的总响应峰值形式可表示为:

总响应峰值 = Σ i R n 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabw gacaqGHbGaae4AaiaabccacaqGsbGaaeyzaiaabohacaqGWbGaae4B aiaab6gacaqGZbGaaeyzaiaabccacaqGubGaae4BaiaabshacaqGHb GaaeiBaiabg2da9maakaaabaGaeu4Odm1aaSbaaSqaaiaadMgaaeqa aOGaamOuamaaDaaaleaacaWGUbaabaGaaGOmaaaaaeqaaaaa@4D35@
R n = A i n ψ i X MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGUbaabeaakiabg2da9iaadgeadaWgaaWcbaGaamyAaiaa d6gaaeqaaOGaeqiYdK3aaSbaaSqaaiaadMgaaeqaaOGaamiwaaaa@3FA9@
其中,

R n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGUbaabeaaaaa@37ED@
代表每个振动模式在单个自由度下的瞬态响应
A MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
特征向量
ψ MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdKhaaa@37C5@
模态参与因子
X MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36D4@
输入响应谱
此公式会计算所有模式响应平方和的平方根,从而估算出总响应峰值。

完全二次项组合 (CQC)

CQC 方法确实能够解决 SRSS 方法在组合自然频率间隔较近的结构的模态响应时的局限性。使用 CQC 方法的总响应峰值可通过公式求得:

总响应峰值 = i Σ j R i ρ i j R j MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabw gacaqGHbGaae4AaiaabccacaqGsbGaaeyzaiaabohacaqGWbGaae4B aiaab6gacaqGZbGaaeyzaiaabccacaqGubGaae4BaiaabshacaqGHb GaaeiBaiabg2da9maakaaabaWaaabeaeaacqqHJoWudaWgaaWcbaGa amOAaaqabaGccaWGsbWaaSbaaSqaaiaadMgaaeqaaOGaeqyWdi3aaS baaSqaaiaadMgacaWGQbaabeaakiaadkfadaWgaaWcbaGaamOAaaqa baaabaGaamyAaaqab0GaeyyeIuoaaSqabaaaaa@5514@
其中,

R i R j MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakiaadkfadaWgaaWcbaGaamOAaaqabaaaaa@39E4@
分别为 i、j 振动模式的峰值模态响应
ρ i j MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39C0@
每个求和步骤合并的两个模态的模态相关系数
模态相关系数 ρ i j MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39C0@ 的计算公式为:
ρ i j = 8 ξ i ξ j ( ξ i + β j i ξ j ) β j i 1 5 ( 1 β j i 2 ) 2 + 4 ξ i ξ j β j i ( 1 + β j i 2 ) + 4 ( ξ i 2 + ξ j 2 ) β j i 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9maalaaabaGaaGioamaa kaaabaGaeqOVdG3aaSbaaSqaaiaadMgaaeqaaOGaeqOVdG3aaSbaaS qaaiaadQgaaeqaaaqabaGccaGGOaGaeqOVdG3aaSbaaSqaaiaadMga aeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaadQgacaWGPbaabeaaki abe67a4naaBaaaleaacaWGQbaabeaakiaacMcacqaHYoGydaqhaaWc baGaamOAaiaadMgaaeaacaaIXaGaeyyXICTaaGynaaaaaOqaaiaacI cacaaIXaGaeyOeI0IaeqOSdi2aa0baaSqaaiaadQgacaWGPbaabaGa aGOmaaaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI0a GaeqOVdG3aaSbaaSqaaiaadMgaaeqaaOGaeqOVdG3aaSbaaSqaaiaa dQgaaeqaaOGaeqOSdi2aaSbaaSqaaiaadQgacaWGPbaabeaakiaacI cacaaIXaGaey4kaSIaeqOSdi2aa0baaSqaaiaadQgacaWGPbaabaGa aGOmaaaakiaacMcacqGHRaWkcaaI0aGaaiikaiabe67a4naaDaaale aacaWGPbaabaGaaGOmaaaakiabgUcaRiabe67a4naaDaaaleaacaWG QbaabaGaaGOmaaaakiaacMcacqaHYoGydaqhaaWcbaGaamOAaiaadM gaaeaacaaIYaaaaaaaaaa@7F32@
其中,
β j i MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadQgacaWGPbaabeaaaaa@39A1@
i 和 j 模态的固有频率之比 ( λ j / λ i MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaadQgaaeqaaOGaai4laiabeU7aSnaaBaaaleaacaWGPbaa beaaaaa@3C51@ )
ξ i MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaadMgaaeqaaaaa@38D4@ ξ j MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiaadQgaaeqaaaaa@38D5@
两个模态的模态阻尼值
此方程考虑了阻尼比、模态固有频率及其相关性之间的关系,与 SRSS 方法相比,可确保更准确地估算总响应峰值。

海军研究实验室 (NRL)

NRL 模态组合方法综合了平方和平方根 (SRSS) 和绝对和 (ABS) 两种方法的优点,实现了一种响应谱分析的平衡方法。它采用 ABS 方法将最大峰值模态响应和其余模态响应相加,计算出的响应具有不同的固有频率,并采用 SRSS 方法进行评估。

NRL 模态组合法的公式如下:

总响应峰值= A ik Ψ i X + ji ( A jk Ψ j X) 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabw gacaqGHbGaae4AaiaabccacaqGubGaae4BaiaabshacaqGHbGaaeiB aiaabccacaqGsbGaaeyzaiaabohacaqGWbGaae4Baiaab6gacaqGZb Gaaeyzaiabg2da9maaemaabaGaamyqamaaBaaaleaacaWGPbGaam4A aaqabaaakiaawEa7caGLiWoadaabdaqaaiabfI6aznaaBaaaleaaca WGPbaabeaakiaadIfaaiaawEa7caGLiWoacqGHRaWkdaGcaaqaamaa qababaGaaiikaiaadgeadaWgaaWcbaGaamOAaiaadUgaaeqaaOGaeu iQdK1aaSbaaSqaaiaadQgaaeqaaOGaamiwaiaacMcadaahaaWcbeqa aiaaikdaaaaabaGaamOAaiabgcMi5kaadMgaaeqaniabggHiLdaale qaaaaa@639F@