JP3640583B2 - Earthquake response analysis method - Google Patents

Description

【００３０】
なお、本実施の形態では、時間積分にはＮｅｗｍａｒｋ−β法を用い、時間積分に用いる剛性には割線剛性を用いている。
【００３１】
次に、第１実施形態の作用として、演算部１６で実行される制御ルーチンについて図２に示すフローチャートを参照して説明する。
【００３２】
まず、オペレータが表示部１８に表示されたメニューに従って操作部１２を操作し、地震応答解析を行うべき解析モデルのパラメータを指定すると、演算部１６では、指定された解析モデルに関するデータを記憶部１４から読み込む。
【００３３】
この解析モデルに関するデータには、例えば解析対象の地層（例えば図３に示すように上から４番目の地層）、材料定数、時間積分定数がある。
【００３４】
次のステップ１０２では、以下の地震応答を求める演算において用いる各種データの初期値の設定を行う。このデータには、例えば加速度、速度、変位、せん断応力、せん断ひずみ、上記（１）式における質量行列Ｍ、剛性行列Ｋ、減衰行列Ｃなどがある。演算部１６では、これらのデータに初期値を設定する。
【００３５】
次に、ステップ１０４では、外力Ｆ（ｔ）の算出を行う。すなわち、上記（１）式の右辺の算出を行う。
【００３６】
そして、次のステップ１０６でひずみと累積損傷度を考慮したせん断剛性Ｇの算出を行う。このせん断剛性Ｇの算出は、せん断応力がゼロ線を横切る毎に行う。すなわち、応答波形の半波毎にせん断剛性Ｇの算出を行う。
【００３７】
せん断剛性Ｇは、せん断ひずみ及び液状化による累積損傷度を考慮したせん断剛性の低下を考慮して次式で示される。
【００３８】
【数２】

【００３９】
ここで、Ｇ₀は、基準平均応力σ^' _m0の時のせん断剛性、γはせん断ひずみ、、Ｄは累積損傷度、γ₅₀はせん断剛性が半分になるせん断ひずみの値で、σ^' _m／σ^' _m0の関数である。現在の有効応力σ^' _mは、応力評価点の初期平均応力σ^' _m0に（１−過剰間隙水圧比＝１−ｒ_u）を掛けた値である。
【００４０】
このように、せん断剛性Ｇは、初期せん断剛性Ｇ₀、せん断ひずみγ、累積損傷度Ｄから求めることができるが、ステップ１０６では、応力時刻歴が前回ゼロ線を横切ってから今回ゼロ線を横切るまでの間（すなわち半波）における最大せん断ひずみγ_max、最大せん断応力τ_maxから、累積損傷度Ｄ、過剰間隙水圧比ｒ_u等を求め、これらからせん断剛性Ｇを求める。
【００４１】
次に、せん断剛性Ｇの算出について、図３を参照して説明する。
【００４２】
本実施の形態では、例えば繰り返し載荷試験等の地盤試験として標準化された試験により得られた図３の▲１▼〜▲４▼に示す関係（式）を用いてせん断剛性Ｇを算出する。
【００４３】
図３に示す▲１▼は、せん断剛性とせん断ひずみとの関係を示すＧ−γ曲線及び減衰とせん断ひずみとの関係を示すｈ−γ曲線を示している。なお、▲１▼の横軸は最大せん断ひずみγ、縦軸はＧ／Ｇ₀（▲１▼において黒丸で示す）及び減衰ｈ（▲１▼において白丸で示す）である。このＧ−γ曲線及びｈ−γ曲線は、地盤調査における動的変形試験により得られたものを用いることができる。
【００４４】
▲１▼に示すように、Ｇ−γ曲線により、せん断ひずみからＧ／Ｇ₀を求めることができ、ｈ−γ曲線から減衰ｈを求めることができる。
【００４５】
図３に示す▲２▼は、液状化強度曲線を示している。なお、▲２▼の横軸は繰り返し回数Ｎ_if、縦軸は応力比τ（せん断応力）／σ_m0である。この液状化強度曲線は、地盤調査における動的非排水試験変形試験により得られたものを用いることができる。
【００４６】
▲２▼に示すように、液状化強度曲線により、せん断応力τから繰り返し回数を求めることができる。さらに、求めた繰り返し回数Ｎ_ifから、半波における累積損傷度の増分ΔＤを求めることができる。半波における累積損傷度の増分ΔＤは、次式で示される。
【００４７】
ΔＤ＝１／２Ｎ_if …（３）
図２に示す▲３▼は、累積損傷度と過剰間隙水圧比との関係を示している。なお、▲３▼の横軸は累積損傷度Ｄ、縦軸は過剰間隙水圧比ｒ_uを示す。この累積損傷度と過剰間隙水圧比との関係は、実験により得ることができるが、公知の次式を用いることができる。
【００４８】
【数３】

【００４９】
図３に示す▲４▼は、過剰間隙水圧比と剛性低下率Ｇ₀との関係を示している。なお、▲４▼の横軸は過剰間隙水圧比ｒ_u、縦軸は剛性低下率Ｇ₀である。この過剰間隙水圧比と剛性低下率Ｇ₀との関係は、実験により得ることができるが、公知の次式を用いることができる。
【００５０】
【数４】

【００５１】
ここで、Ｇ_refはσ_refでのせん断剛性である。また、過剰間隙水圧比とγ₅₀との関係には次式を用いることができる。
【００５２】
【数５】

【００５３】
ここで、（γ₅₀）_refはσ_refでのγ₅₀、γ₅₀はＧ／Ｇ₀が０．５となるせん断ひずみである。また、前述したように、現在の有効応力σ^' _mは、応力評価点の初期平均応力σ^' _m0に（１−過剰間隙水圧比＝１−ｒ_u）を掛けた値である。
【００５４】
ステップ１０６では、これらの関係を用いてせん断剛性Ｇを算出する。すなわち、後述するステップ１２２において前回算出した最大せん断ひずみγ_max、後述するステップ１２２において前回算出した最大せん断応力τ_maxから上記▲２▼、▲３▼、▲４▼の関係（すなわち式（３）〜（６））を用いて得られた剛性低下率Ｇ₀、累積損傷度Ｄを式（２）へ代入し、せん断剛性Ｇを求める。
【００５５】
次に、ステップ１０８において、要素剛性行列の算出と全体剛性行列の組み立てを行う。すなわち、（１）式における剛性行列Ｋを算出する。
【００５６】
次に、ステップ１１０において、減衰行列Ｃを算出する。減衰行列Ｃは、図３の▲５▼に示すように次式で示すレイリー減衰を用いる。
【００５７】
Ｃ＝αＭ＋βＫ …（７）
ここで、α、βはω₁、ω₂、ｈから定まる係数である。なお、ω₁、ω₂は、せん断剛性Ｇから定まる固有値である。なお、減衰行列Ｃは、レイリー減衰に限らず他の方法を用いてもよい。
【００５８】
従って、ステップ１１０では、図３の▲１▼のｈ−γ曲線を用いて最大せん断ひずみγ_maxから対応する減衰ｈを求め、この減衰ｈから減衰行列Ｃを算出する。
【００５９】
次に、ステップ１１２において、動的解析用全体行列Ｋ^*を算出すると共に、そのＬＵ分解を行う。動的解析用全体行列Ｋ^*は次式で示される。
【００６０】
Ｋ^*＝Ｍ＋γΔｔＣ＋βΔｔ²Ｋ …（８）
ここで、β、γは、Ｎｅｗｍａｒｋ−β法における係数（一定）である。
【００６１】
次に、ステップ１１４において、ｔ＋Δｔ時刻の予測値の算出と、該算出した予測値に対する内外力の算出を行う。Δｔは計算時間間隔、すなわち時刻歴ループの実行間隔である。なお、Δｔは一定の値でもよいし、計算毎に変化させてもよい。予測値としては、外挿による変位及び速度を求める。そして、求めた予測値から内外力Ｒ（ｔ＋Δｔ）を求める。内外力Ｒ（ｔ＋Δｔ）は次式で示される。
【００６２】
【数６】

【００６３】
次に、ステップ１１６において、求めた内外力Ｒから不釣合い力ΔＲ（＝Ｆ−Ｒ）を求める。
【００６４】
次に、ステップ１１８において、修正値Δｕの計算を行う。修正値Δｕは次式で示される。
【００６５】
【数７】

【００６６】
このΔｕから変位、速度、加速度をそれぞれ修正する。
【００６７】
次に、ステップ１２０において、ｔ＋Δｔ時刻の変位に基づくせん断応力τ、せん断ひずみγを算出する。
【００６８】
次に、ステップ１２２において、半波毎の最大せん断応力τ_max及び最大せん断ひずみγ_maxを求める。すなわち、ステップ１２０で求めたせん断応力τと現在の最大せん断応力τ_maxとを比較し、せん断応力τが最大せん断応力τ_maxよりも大きい場合には、最大せん断応力τ_maxをせん断応力τで書き換える。同様に、ステップ１２０で求めたせん断ひずみγと最大せん断ひずみγ_maxとを比較し、せん断ひずみγが最大せん断ひずみγ_maxよりも大きい場合には、最大せん断ひずみγ_maxをせん断ひずみγで書き換える。そして、せん断応力がゼロ線を横切る毎に最大せん断応力τ_max及び最大せん断ひずみγ_maxを記憶部１４へ記憶し、最大せん断応力τ_max及び最大せん断ひずみγ_maxを初期化する。上記の処理を繰り返すことにより、半波毎の最大せん断応力τ_max及び最大せん断ひずみγ_maxを求めることができる。
【００６９】
次に、ステップ１２４において、算出した各応答値を表示部１８へ出力する。出力値としては、変位、速度、加速度、せん断ひずみ、せん断応力等がある。
【００７０】
このようにして、地震時間全体についてステップ１０４からステップ１２４までの処理（時刻歴ループ）を行い、各時刻毎の応答値を算出する。なお、ステップ１０６、１０８、１１０、１１２は、せん断応力がゼロ線を横切る毎に行えばよいため、各時刻の応答値を算出する毎に毎回行う必要はない。
【００７１】
このようにして時刻歴ループが終了すると、ステップ１２６において、時刻歴ループにおいて算出した各応答値の最大値などを出力する。
【００７２】
このように、地震応答の半波毎に、地盤材料試験などから得られたデータ（経験式）から簡単に得られる累積損傷度により液状化を考慮した地盤剛性を求め、これを用いて時刻歴応答計算を行うので、解析するものの技量によらない地震応答及び永久変形の予測を行うことが可能になると共に、地震応答の予測技術の標準化が進み、液状化時の建設構造物の安全性が客観的に評価することが可能となる。また、半波毎に割線剛性を用いた線形計算を用いるので運動方程式の解を安定して求めることができ、アルゴリズムによる誤差の少ない地震応答を得ることができる。
【００７３】
なお、上記では、時刻歴応答計算を行うのに半波毎の最大値を用いたが、これに限らず、半波毎の平均値、最大応答値に所定の係数を掛けけたものを用いて時刻歴応答計算を行うようにしてもよい。
【００７４】
〔第２実施形態〕
次に、本発明の第２実施形態について説明する。
【００７５】
第１実施形態では、１つ前の半波の応答値に対応する地盤剛性を用いて次の半波の応答を求めた。すなわち、１つ前の半波の最大せん断ひずみγ_max及び最大せん断応力τ_maxからせん断剛性Ｇを求めて時刻歴応答計算を行っていた。しかし、この場合、現在の半波のせん断剛性Ｇを求める際に１つ前の半波の応答値から地盤剛性を求めることになるため、大地震の時のように、急に地震振幅が大きくなると１つ前の半波と今回の半波の応答が大きく異なる場合があり、実際の地盤剛性と計算による地盤剛性との差が大きくなる場合がある。第２実施形態では、これを解決するために、地震応答解析を繰り返して行い、１回目に求めたせん断ひずみと累積損傷度を用いて次回以降の地震応答解析を行うことにより実際の地盤剛性と計算による地盤剛性との誤差を小さくする方法について説明する。
【００７６】
以下、第２実施形態における作用として、演算部１６で実行される制御ルーチンについて図４に示すフローチャートを参照して説明する。
【００７７】
まず、ステップ２００では、前述した図２に示す制御ルーチンを実行し、１回目の地震応答解析を行う。なお、このとき、図２に示すステップ１０６及びステップ１２２で、図５に示すように半波毎に求めた累積損傷度Ｄ及び最大せん断ひずみγ_maxを記憶部１４へ逐次記憶しておく。すなわち、累積損傷度Ｄ及び最大せん断ひずみγ_maxの時刻歴等を記憶しておく。また、各応答値（加速度等）の最大値等を記憶しておいてもよい。
【００７８】
そして、次のステップ２０２では、ステップ２００で記憶しておいた累積損傷度Ｄ及び最大せん断ひずみγ_maxの時刻歴を用いて、図２に示す制御ルーチンを実行し、２回目の地震応答解析を行う。すなわち、２回目の地震応答解析では、図２に示すステップ１０６において、現在の半波のせん断剛性Ｇを、１つ前の半波における累積損傷度Ｄ及び最大せん断ひずみγ_maxから求めるのではなく、図６に示すように１回目（前回）の地震応答解析により求めた同時刻又はこの時刻より前で最も近い最大せん断ひずみγ_maxとその時刻の累積損傷度Ｄから求める。そして、求めたせん断剛性Ｇから図７に示すように、次回の地震応答解析のための最大せん断ひずみγ_max及び最大せん断応力τ_maxを求める。
【００７９】
次に、ステップ２０４において、１回目の地震応答解析において得られた例えば最大せん断ひずみγ_maxの時刻歴と２回目の地震応答解析により得られた最大せん断ひずみγ_maxの時刻歴とを比較する。そして、両者の時刻歴が略一致した場合には、１回目の地震応答解析が正しかったと判断して、次のステップ２０６で応答値などを出力して終了する。
【００８０】
なお、略一致したか否かの判断は、例えば各時刻毎に１回目の地震応答解析において得られた最大せん断ひずみγ_maxと２回目の地震応答解析により得られた最大せん断ひずみγ_maxとをそれぞれ比較し、各時刻における両者の差が全て又は所定以上において許容値以内であれば両者の時刻歴が略一致したと判断することができる。また、ステップ２０４における比較は、累積損傷度Ｄ、加速度、速度、変位等の時刻歴やこれらの全時刻における最大値等を用いて行ってもよい。
【００８１】
一方、１回目の地震応答解析において得られた例えば最大せん断ひずみγ_maxの時刻歴と２回目の地震応答解析により得られた最大せん断ひずみγ_maxの時刻歴とが略一致しなかった場合には、ステップ２０２へ戻って３回目の地震応答解析を行う。このとき、せん断剛性Ｇは、２回目の地震応答解析により求めた同時刻における累積損傷度Ｄ及び最大せん断ひずみγ_maxから求める。
【００８２】
このように、せん断剛性Ｇを、図４及び図８に示すように前回の地震応答解析により求めた同時刻における最大せん断ひずみγ_max及び累積損傷度Ｄから求め、前回求めた最大せん断ひずみγ_maxの時刻歴と今回求めた最大せん断ひずみγ_maxの時刻歴とが略一致するまで地震応答解析を繰り返し行う。これにより、大地震のときのように急に地震振幅が大きくなるような場合でも地盤剛性の誤差を小さくすることができ、精度よく地震応答解析を行うことができる。
【００８３】
【実施例】
次に、本発明の実施例について図面を参照して説明する。本実施例では、第１の実施形態で説明した地震応答解析方法の実施例について説明する。
【００８４】
以下の表１に示す解析ケースについて第１実施形態で示した地震応答解析装置１０により地震応答解析を行い、これらの解析結果と線形応答解析、時刻歴非線形解析、動的有効応力解析による解析結果との比較を行った。なお、解析プログラムには、動的有効応力解析を含む非線形解析にはＭｕＤＩＡＮを、線形解析及び等価線形解析にはＳＨＡＫＥを用いた。この結果、線形解析については全く同じ解を得、等価線形解析についてもかなり近い解を得ることができた。
【００８５】
【表１】

【００８６】
次に、Ｒａｍｂｅｒｇ−Ｏｓｇｏｏｄ（Ｒ−Ｏ）モデルによる非線形応答解析と、Ｄｅｎｓｉｆｉｃａｔｉｏｎモデルを用いた有効応力解析との比較結果について説明する。
【００８７】
検討に用いた地盤モデルは深さ約２１．４ｍの迄が液状化の可能性のある実地盤で、解析には−５０ｍ迄のモデル化を行った。地盤定数を表２に示す。標準貫入試験は、−２６．３ｍまでであり、それ以深−４０ｍまでは、−２６．３ｍとと同じ物性としている。また、図３に示すＧ−γ曲線、ｈ−γ曲線、及び液状化強度曲線には、凍結サンプリングによる−４ｍ、−６ｍ、−９ｍ、−１２ｍ、−１５ｍにおける液状化試験、動的変形試験の結果を用いた。図９にＧ／Ｇ₀−γ曲線及び液状化強度曲線を示す。また、Ｒ−Ｏモデル、Ｄｅｎｓｉｆｉｃａｔｉｏｎモデルのパラメータを表３に示す。なお、−２１．４ｍ以深は線形材料とし、ｈ＝２％とした。
【００８８】
【表２】

【００８９】
【表３】

【００９０】
まず、Ｒ−Ｏモデルによる解析結果との比較を示す。図１０に表層（Ｇ．Ｌ．０．０ｍｍ：Ｌａｙｅｒ１）とＧ．Ｌ．−１４．６５ｍ（Ｌａｙｅｒ６−７）での応答加速度時刻歴を示す。なお、太線が本発明の地震応答解析方法（ＬＥＱ）での解析結果（解析ケースは表１におけるＥ−Ｒ２）であり、細線がＲ−Ｏモデルでの解析結果である。また、図１１に最大応答加速度分布図及び最大せん断応力分布図を示す。
【００９１】
図１０に示すように、波形はそれぞれの振幅が良く対応しているが、Ｒ−Ｏモデルでは高振動成分があり、ピーク値に差が生じている。これが図１１に示す最大応答加速度分布にも反映されており、Ｇ．Ｌ．−１０ｍ〜−２５ｍでは大きく、それ以外では小さくなっているが全体では良く一致している。
【００９２】
次に、Ｄｅｎｓｉｆｉｃａｔｉｏｎモデル（有効応力解析）による解析結果との比較について説明する。
【００９３】
図１２に液状化率を示す。上から順に、Ｌａｙｅｒ２（Ｇ．Ｌ．−２．３ｍ〜−４．４ｍ）、Ｌａｙｅｒ５（Ｇ．Ｌ．−９．７ｍ〜−１２．５ｍ）、Ｌａｙｅｒ７（Ｇ．Ｌ．−１４．６５ｍ〜−１８．６ｍ）の結果を示す。なお、太線が本発明の地震応答解析方法（ＬＥＱ）での解析結果（解析ケースは表１におけるＡ−Ｒ１）であり、細線がＤｅｎｓｉｆｉｃａｔｉｏｎモデルでの解析結果である。また、図１３に応答加速度時刻歴を、図１４に最大応答加速度分布及び最大応答せん断応力分布図を示す。
【００９４】
図１２に示すように、本発明による地震応答解析方法では、半波毎に液状化率（過剰間隙水圧比）を計算するため階段状となっているが、全体的にはＤｅｎｓｉｆｉｃａｔｉｏｎモデルによる解析結果と良い対応をしている。
【００９５】
これによると、表層直下で液状化率が約０．７となり剛性が大きく低下していることが予想される。これを反映して、図１３に示す応答加速度時刻歴は、１０秒近くで応答が小さくなると共に長周期化している。また、中間層（Ｇ．Ｌ．−１４．６５ｍ：Ｌａｙｅｒ６−７）は、１５秒を過ぎて液状化していることが図１２に示した液状化率及び図１３の応答加速度時刻歴から認識することができる。この結果は、Ｄｅｎｓｉｆｉｃａｔｉｏｎモデルによる解析結果と良く一致している。
【００９６】
最大応答加速度及び最大応答せん断応力は、液状化を考慮しないＲ−Ｏモデルでの非線形解析による解析結果の場合よりやや差が大きいが、全体としては良い対応をしているといえる。
【００９７】
【発明の効果】
以上説明したように、本発明によれば、解析するものの技量によらない地震応答及び永久変形の予測を行うことが可能になると共に、地震応答の予測技術の標準化が進み、液状化時の建設構造物の安全性が客観的に評価することが可能となると共に、大地震のときのように急に地震振幅が大きくなるような場合でも地盤剛性の誤差を小さくすることができ、精度よく地震応答解析を行うことができる。また、本発明は、地盤だけでなく骨組み構造にも適用できると共に計算機が高速になれば一般構造物にも適用可能であり、非線形動的問題の安定した解析法として有用である。
【図面の簡単な説明】
【図１】 地震応答解析装置の概略ブロック図である。
【図２】 第１実施形態における演算部において実行される制御ルーチンのフローチャートである。
【図３】 せん断剛性の算出について説明するための図である。
【図４】 第２実施形態における演算部において実行される制御ルーチンのフローチャートである。
【図５】 第２実施形態におけるせん断ひずみ、累積損傷度の第１回目の計算結果について説明するための図である。
【図６】 第２実施形態におけるせん断剛性の第２回目以降の算出について説明するための図である。
【図７】 第２実施形態における次回のための準備計算について説明するための図である。
【図８】 第２実施形態におけるせん断剛性の算出について説明するための図である。
【図９】 （ａ）はＧ−γ曲線を示す線図、（ｂ）は液状化強度曲線の線図である。
【図１０】 応答加速度時刻歴を示す線図である。
【図１１】 最大応答加速度分布及び最大せん断応力分布を示す線図である。
【図１２】 液状化率の時刻歴を示す線図である。
【図１３】 応答加速度時刻歴を示す線図である。
【図１４】 最大応答加速度分布及び最大せん断応力分布を示す線図である。
【符号の説明】
１０ 地震応答解析装置
１２ 操作部
１４ 記憶部
１６ 演算部
１８ 表示部[0001]
BACKGROUND OF THE INVENTION
The present invention relates to an earthquake response analysis method, and more particularly, to an earthquake response analysis method that takes into account changes in ground rigidity.
[0002]
[Prior art]
Conventionally, earthquake response analysis methods that take into account changes in ground stiffness during an earthquake include methods that take into account changes in tangential stiffness at each calculation step and linear response analysis when calculating the response during an earthquake by time integration. Repeat for the entire earthquake time, obtain the average stiffness that has decreased over the entire time from the previous analysis result, that is, the equivalent linear stiffness, and use it for the next calculation until the equivalent linear stiffness is constant There are methods.
[0003]
However, with the former method, it is difficult to directly obtain a coefficient for determining a method (stress-strain relationship or constitutive equation) for obtaining a change in ground rigidity from a ground constant. In addition, for a random phenomenon such as an earthquake, the method of following in detail every moment is not standardized, and the analysis results vary greatly depending on the analysis method.
[0004]
In the latter method, when the ground stiffness changes greatly during an earthquake (for example, liquefaction), the response varies greatly before and after that, so calculation is performed using one ground stiffness throughout the entire earthquake time. If so, a large error occurs.
[0005]
[Problems to be solved by the invention]
Japanese Patent No. 2742798 discloses a technique for forcibly generating vibration in the formation, measuring changes in physical properties of the ground, and determining the ease of liquefaction.
[0006]
However, this technique predicts the strength of liquefaction rather than judging liquefaction, and cannot predict how liquefaction will occur during an actual earthquake.
[0007]
Japanese Patent No. 2743975 discloses a technique for measuring the fine movement of the ground and determining liquefaction based on the measurement result.
[0008]
However, since this technique does not take into account the magnitude of the input seismic motion, liquefaction can be determined only roughly, and prediction of the liquefaction occurrence time cannot be accurately determined.
[0009]
The present invention has been made to solve the above problems, and an object of the present invention is to provide an earthquake response analysis method capable of obtaining an earthquake response with little error in consideration of changes in liquefaction and ground rigidity. .
[0010]
[Means for Solving the Problems]
Liquefaction occurs by causing volume change (dilatancy) by sand undergoing shear deformation. Therefore, in order to obtain the earthquake response at the time of liquefaction, it is necessary to know the relationship between shear deformation and volume change, and to incorporate that relationship into the ground equation of motion. The former relationship includes the concept of cumulative damage.
[0011]
  Therefore, the invention according to claim 1 is the mass of the model to be analyzed.matrixInformation, rigiditymatrixInformation, decaymatrixInformation and displacement information of the model to be analyzed that changes moment by moment using an equation of motion based on input earthquake information, and based on the displacement information, at least a strain response value that changes moment by momentAnd stress response valuesIn the seismic response analysis method to obtain the time characteristics ofBy time integrating the equation of motionBased on the obtained displacement information, find the response value that changes from moment to moment,Within a half wave from when the response value related to the strain becomes a predetermined response value to the next predetermined response valueSoughtRegarding the strainResponse valueOf the representative value and the response value for the stressObtained a representative valueTypical values for stress and stressRigidity based on typical valuesAnd decayAnd obtained rigidityAnd decayIncluding the rigiditymatrixinformationAnd the attenuation matrix informationTheHalf waveIt is characterized by updating every time.
[0012]
  According to the present invention, the mass obtained from the material information etc. of the analysis target model obtained from the material test etc.matrixInformation, rigiditymatrixInformation, decaymatrixDisplacement information of the model to be analyzed that changes momentarily from the equation of motion based on the information and the input seismic information obtained by measuring actual earthquakes is obtained. this is,luckThe dynamic equation can be obtained by time integration. As described in claim 4, the analysis target model can be a ground model or a model of a frame structure such as a building.
[0013]
  AndBy integrating the equation of motion with timeResponse value for at least strain (eg shear strain) that changes from moment to moment based on the displacement information obtainedAnd stress response valuesFind the time characteristics of. The response value includes strainAnd stressBesides, for example, speed, acceleration,CumulativeThere is a degree of product damage, a plasticity rate, etc.
[0014]
  In obtaining the time characteristic of such a response value,Response value related to strain is a predetermined response valueFor example, almost zeroThe representative value of the response value related to the strain and the representative value of the response value related to the stress obtained in the half wave from when the value reaches the next predetermined response value are obtained.In other words, the representative valueResponse valueThis time after becoming almost zeroResponse valueUntil is almost zeroIn the half waveThis is a representative value of the obtained response value. This representative value is, for example, the last timeResponse valueThis time after becoming almost zeroResponse valueThe maximum or average response value obtained until the value becomes substantially zero, or a value obtained by multiplying these by a predetermined coefficient..
[0015]
  In this way, at predetermined intervalsTypical values of response values related to strain and response values related to stressObtain a representative value, and stiffness (for example, shear stiffness) based on the obtained representative valueAnd decayAsk for. This stiffness is obtained by, for example, the Young's modulus, Poisson's ratio, bulk modulus, shear modulus, etc. being interrelated, and can be obtained by the secant stiffness method.
[0016]
  And the calculated rigidityAnd decayIncluding rigidmatrixinformationAnd attenuation matrix informationTheHalf waveUpdate every time. That is, the calculated rigidityAnd decayIs related to the equation of motionmatrixinformationAnd attenuation matrix informationAs reflected inHalfUpdate every wave.
[0017]
By repeating the above process, the time characteristic of the response value, that is, the time history is obtained. Thus, since the time history of the response value is obtained by performing a linear analysis for each half wave, the solution of the equation of motion can be obtained stably, and an earthquake response with less error can be obtained.
[0018]
Note that, as described in claim 2, the rigidity may be obtained based on a representative value obtained immediately before (that is, before half a wave). In this case, since the rigidity is obtained based on the representative value before the half wave, the response of the previous half wave and the current half wave may be greatly different if the earthquake amplitude suddenly increases, as in the case of a large earthquake. Yes, the difference between the actual stiffness and the calculated stiffness may be large.
[0019]
  Therefore, as described in claim 3, after obtaining the time characteristic of the response value,By time integrating the equation of motionBased on the obtained displacement information, find a response value that changes from moment to moment,Within a half wave from when the response value related to the strain becomes a predetermined response value to the next predetermined response valueSoughtRegarding the strainObtain a representative value of the response value,StrainLast time corresponding to the time when response value was calculatedRegarding the strainRigidity based on typical valuesAnd decayAnd obtained rigidityAnd decayIncluding the rigiditymatrixinformationAnd the attenuation matrix informationTheHalf waveThe time characteristic is obtained by repeating the update every time. If the time characteristic obtained last time and the time characteristic obtained this time are not substantially matched, the time characteristic obtained last time and the time characteristic obtained this time are substantially the same. Until,By time integrating the equation of motionBased on the obtained displacement information, find a response value that changes from moment to moment,Within the half waveSoughtRegarding the strainObtain a representative value of the response valueStrainLast time corresponding to the time when response value was calculatedRegarding the strainRigidity based on typical valuesAnd decayAnd obtained rigidityAnd decayIncluding the rigiditymatrixinformationAnd the attenuation matrix informationTheHalf waveIt is also possible to obtain the time characteristic by repeating the update every time, and repeat the process of comparing the time characteristic obtained last time with the time characteristic obtained this time.
[0020]
  That is, after obtaining the time characteristic of the first response value, the same as the case of obtaining the first time characteristic.Based on the displacement information obtained by time integration of the equation of motion, a response value that changes every moment is obtained, and in the half wave from when the response value related to strain becomes the predetermined response value to the next predetermined response value Response value for the obtained strainObtain a representative value.
  And this time, the rigidity was calculated just before (that is, half wave)StrainRather than calculating based on representative valuesStrainOf the representative values, this timeStrainLast time corresponding to the time when response value was calculatedStrainRigidity based on typical valuesAnd decayAsk for. And the calculated rigidityAnd decayIncluding rigidmatrixinformationAnd attenuation matrix informationTheHalfEvery waveInupdateYouThe As a result, there is no time mismatch, and even when the earthquake amplitude suddenly increases as in the case of a large earthquake, the rigidity error can be reduced, and the seismic response analysis can be performed with high accuracy.
[0021]
  When the analysis target model is a ground model, it may be necessary to perform an earthquake response analysis in consideration of liquefaction due to an earthquake. Therefore, as described in claim 5,DemandStressConcerningCalculate the cumulative damage from the representative value,About stressThe stiffness may be obtained based on the response value and the cumulative damage level.
[0022]
  In this way, the stress obtained for each half waveConcerningBy obtaining the cumulative damage degree from the representative value and obtaining the rigidity based on the representative value such as shear strain obtained for each half wave and the obtained cumulative damage degree, the seismic response analysis considering liquefaction becomes possible. In this way, the ground stiffness considering liquefaction is calculated for each half-wave according to the cumulative damage degree, and the time history response calculation is performed using this, so the seismic response and permanent deformation are predicted regardless of the skill of analysis. As a result, the standardization of earthquake response prediction technology advances, and the safety of construction structures during liquefaction can be objectively evaluated. In addition, by performing a linear analysis for each half wave, the solution of the equation of motion can be obtained stably, and an earthquake response with few errors can be obtained.
[0023]
DETAILED DESCRIPTION OF THE INVENTION
[First Embodiment]
Hereinafter, a first embodiment of the present invention will be described with reference to the drawings.
[0024]
FIG. 1 shows an earthquake response analyzing apparatus 10. The earthquake response analysis apparatus 10 includes an operation unit 12, a storage unit 14, a calculation unit 16, and a display unit 18.
[0025]
The operation unit 12 is for an operator to specify an instruction and necessary parameters for causing the arithmetic unit 16 to execute an earthquake response analysis for a desired analysis model according to a menu displayed on the display unit 18. The storage unit 18 stores various analysis model parameters and various calculation expressions necessary for earthquake response analysis in the calculation unit 16. In addition, the storage unit 16 stores an analysis result of the earthquake response analysis by the calculation unit 16.
[0026]
The calculation unit 16 reads necessary data from the storage unit 18 according to an instruction from the operation unit 12 and performs an earthquake response analysis, stores the output result in the storage unit 16, and outputs it to the display unit 18.
[0027]
The calculation unit 16 performs an earthquake response analysis in the following manner. That is, the cumulative damage degree is obtained from the time history of the shear stress, the excess pore water pressure is obtained from the cumulative damage degree, and the shear stiffness reduction rate due to liquefaction is obtained from the effective stress obtained thereby. Then, for each half wave of the amplitude of shear stress and shear strain, the stiffness reduction rate due to the maximum shear strain in the half wave is obtained, and from this, the equivalent shear stiffness is obtained, and this stiffness (secant stiffness) is used. Time history response analysis, that is, earthquake response analysis is performed.
[0028]
Earthquake response analysis is performed by time integration of the equation of motion shown by the following equation.
[0029]
[Expression 1]

[0030]
In the present embodiment, the Newmark-β method is used for time integration, and secant rigidity is used for rigidity used for time integration.
[0031]
Next, as an operation of the first embodiment, a control routine executed by the calculation unit 16 will be described with reference to a flowchart shown in FIG.
[0032]
First, when the operator operates the operation unit 12 according to the menu displayed on the display unit 18 and specifies the parameters of the analysis model to be subjected to the earthquake response analysis, the calculation unit 16 stores the data related to the specified analysis model in the storage unit 14. Read from.
[0033]
The data relating to the analysis model includes, for example, a formation to be analyzed (for example, the fourth formation from the top as shown in FIG. 3), a material constant, and a time integration constant.
[0034]
In the next step 102, initial values of various data used in the calculation for obtaining the following earthquake response are set. This data includes, for example, acceleration, velocity, displacement, shear stress, shear strain, mass matrix M, stiffness matrix K, and attenuation matrix C in the above equation (1). The calculation unit 16 sets initial values for these data.
[0035]
Next, in step 104, the external force F (t) is calculated. That is, the right side of the above equation (1) is calculated.
[0036]
In the next step 106, the shear rigidity G is calculated in consideration of the strain and the cumulative damage degree. The calculation of the shear stiffness G is performed every time the shear stress crosses the zero line. That is, the shear stiffness G is calculated for each half wave of the response waveform.
[0037]
The shear stiffness G is expressed by the following equation in consideration of a decrease in shear stiffness in consideration of the shear strain and the cumulative damage due to liquefaction.
[0038]
[Expression 2]

[0039]
Where G₀Is the standard mean stress σ^' _m0Shear stiffness, γ is shear strain, D is cumulative damage, γ₅₀Is the shear strain value at which the shear stiffness is halved.^' _m/ Σ^' _m0Is a function of Current effective stress σ^' _mIs the initial mean stress σ of the stress evaluation point^' _m0(1-excess pore water pressure ratio = 1-r_u).
[0040]
Thus, the shear stiffness G is equal to the initial shear stiffness G.₀, Shear strain γ, and cumulative damage degree D, but in step 106, the maximum shear strain γ from the time when the stress time history crosses the previous zero line to the current zero line (ie, half wave)._maxMaximum shear stress τ_maxFrom cumulative damage D, excess pore water pressure ratio r_uEtc., and the shear rigidity G is obtained from these.
[0041]
Next, calculation of the shear stiffness G will be described with reference to FIG.
[0042]
In the present embodiment, for example, the shear stiffness G is calculated using the relationships (formulas) shown in (1) to (4) in FIG. 3 obtained by a standardized test as a ground test such as a repeated loading test.
[0043]
(1) shown in FIG. 3 shows a G-γ curve showing the relationship between shear stiffness and shear strain and an h-γ curve showing the relationship between damping and shear strain. The horizontal axis of (1) is the maximum shear strain γ, and the vertical axis is G / G.₀(Indicated by black circles in (1)) and attenuation h (indicated by white circles in (1)). As this G-γ curve and h-γ curve, those obtained by a dynamic deformation test in ground investigation can be used.
[0044]
As shown in (1), the G / G curve shows the shear strain to G / G₀And attenuation h can be obtained from the h-γ curve.
[0045]
(2) shown in FIG. 3 indicates a liquefaction strength curve. The horizontal axis of (2) is the number of repetitions N_if, Vertical axis is stress ratio τ (shear stress) / σ_m0It is. As the liquefaction strength curve, a curve obtained by a dynamic undrained test deformation test in the ground survey can be used.
[0046]
As shown in (2), the number of repetitions can be determined from the shear stress τ using the liquefaction strength curve. Furthermore, the obtained number of repetitions N_ifFrom the above, the increment ΔD of the cumulative damage degree in the half wave can be obtained. The cumulative damage degree increment ΔD in the half-wave is expressed by the following equation.
[0047]
ΔD = 1 / 2N_if    ... (3)
(3) shown in FIG. 2 indicates the relationship between the cumulative damage degree and the excess pore water pressure ratio. The horizontal axis of (3) is the cumulative damage degree D, and the vertical axis is the excess pore water pressure ratio r._uIndicates. The relationship between the cumulative damage degree and the excess pore water pressure ratio can be obtained by experiments, but the following known equation can be used.
[0048]
[Equation 3]

[0049]
(4) shown in FIG. 3 indicates the excess pore water pressure ratio and the rigidity reduction rate G.₀Shows the relationship. The horizontal axis of (4) is the excess pore water pressure ratio r._u, The vertical axis is the stiffness reduction rate G₀It is. This excess pore water pressure ratio and stiffness reduction rate G₀The relationship between and can be obtained by experiment, but the following well-known formula can be used.
[0050]
[Expression 4]

[0051]
Where G_refIs σ_refShear stiffness at Also, excess pore water pressure ratio and γ₅₀The following equation can be used for the relationship:
[0052]
[Equation 5]

[0053]
Where (γ₅₀)_refIs σ_refΓ in₅₀, Γ₅₀Is G / G₀Is a shear strain at which is 0.5. In addition, as described above, the current effective stress σ^' _mIs the initial mean stress σ of the stress evaluation point^' _m0(1-excess pore water pressure ratio = 1-r_u).
[0054]
In step 106, the shear stiffness G is calculated using these relationships. That is, the maximum shear strain γ previously calculated in step 122, which will be described later._maxThe maximum shear stress τ previously calculated in step 122 described later_maxFrom the above-mentioned relationships (2), (3), and (4) (that is, the equations (3) to (6)), the rigidity reduction rate G₀Then, the cumulative damage degree D is substituted into the equation (2), and the shear rigidity G is obtained.
[0055]
Next, in step 108, the element stiffness matrix is calculated and the entire stiffness matrix is assembled. That is, the stiffness matrix K in equation (1) is calculated.
[0056]
Next, in step 110, an attenuation matrix C is calculated. As the attenuation matrix C, Rayleigh attenuation represented by the following equation is used as indicated by (5) in FIG.
[0057]
C = αM + βK (7)
Where α and β are ω₁, Ω₂, H is a coefficient determined from h. Ω₁, Ω₂Is an eigenvalue determined from the shear stiffness G. The attenuation matrix C is not limited to Rayleigh attenuation, and other methods may be used.
[0058]
Therefore, in step 110, the maximum shear strain γ is obtained using the h-γ curve of (1) in FIG._maxThe corresponding attenuation h is obtained from the above, and the attenuation matrix C is calculated from the attenuation h.
[0059]
Next, in step 112, the dynamic analysis whole matrix K^*And LU decomposition is performed. Overall matrix K for dynamic analysis^*Is expressed by the following equation.
[0060]
K^*= M + γΔtC + βΔt²K ... (8)
Here, β and γ are coefficients (constant) in the Newmark-β method.
[0061]
Next, in step 114, the predicted value at time t + Δt is calculated, and the internal / external force for the calculated predicted value is calculated. Δt is a calculation time interval, that is, a time history loop execution interval. Note that Δt may be a constant value or may be changed for each calculation. As the predicted value, the displacement and speed by extrapolation are obtained. Then, the internal / external force R (t + Δt) is obtained from the obtained predicted value. The internal / external force R (t + Δt) is expressed by the following equation.
[0062]
[Formula 6]

[0063]
Next, in step 116, an unbalanced force ΔR (= F−R) is obtained from the obtained internal / external force R.
[0064]
Next, in step 118, a correction value Δu is calculated. The correction value Δu is expressed by the following equation.
[0065]
[Expression 7]

[0066]
The displacement, speed, and acceleration are corrected from this Δu.
[0067]
Next, in step 120, a shear stress τ and a shear strain γ based on the displacement at time t + Δt are calculated.
[0068]
Next, in step 122, the maximum shear stress τ for each half wave_maxAnd maximum shear strain γ_maxAsk for. That is, the shear stress τ obtained in step 120 and the current maximum shear stress τ_maxAnd the shear stress τ is the maximum shear stress τ_maxIs greater than the maximum shear stress τ_maxIs rewritten by the shear stress τ. Similarly, the shear strain γ and the maximum shear strain γ determined in step 120_maxAnd the shear strain γ is the maximum shear strain γ_maxIs greater than the maximum shear strain γ_maxIs rewritten by the shear strain γ. And every time the shear stress crosses the zero line, the maximum shear stress τ_maxAnd maximum shear strain γ_maxIs stored in the storage unit 14 and the maximum shear stress τ is stored._maxAnd maximum shear strain γ_maxIs initialized. By repeating the above process, the maximum shear stress τ for each half wave_maxAnd maximum shear strain γ_maxCan be requested.
[0069]
Next, in step 124, the calculated response values are output to the display unit 18. Output values include displacement, speed, acceleration, shear strain, shear stress, and the like.
[0070]
In this way, the processing from Step 104 to Step 124 (time history loop) is performed for the entire earthquake time, and the response value for each time is calculated. Note that steps 106, 108, 110, and 112 need only be performed every time the shear stress crosses the zero line, and therefore need not be performed every time a response value at each time is calculated.
[0071]
When the time history loop is thus completed, in step 126, the maximum value of each response value calculated in the time history loop is output.
[0072]
Thus, for each half wave of the seismic response, the ground stiffness considering liquefaction is obtained from the cumulative damage degree easily obtained from the data (empirical formula) obtained from the ground material test etc. Since response calculation is performed, it is possible to predict seismic response and permanent deformation regardless of the skill of analysis, and standardization of earthquake response prediction technology advances, and the safety of construction structures during liquefaction is improved. Objective evaluation is possible. In addition, since a linear calculation using secant stiffness is used for each half-wave, the solution of the equation of motion can be obtained stably, and an earthquake response with little error by the algorithm can be obtained.
[0073]
In the above, the maximum value for each half wave is used to calculate the time history response, but not limited to this, the average value for each half wave and the maximum response value multiplied by a predetermined coefficient are used. Time history response calculation may be performed.
[0074]
[Second Embodiment]
Next, a second embodiment of the present invention will be described.
[0075]
In the first embodiment, the response of the next half wave is obtained using the ground rigidity corresponding to the response value of the previous half wave. That is, the maximum shear strain γ of the previous half wave_maxAnd maximum shear stress τ_maxThe shear history G was obtained from the time history response calculation. However, in this case, since the ground stiffness is obtained from the response value of the previous half wave when obtaining the current half wave shear stiffness G, the earthquake amplitude suddenly increases as in the case of a large earthquake. Then, the response of the previous half-wave and the current half-wave may be greatly different, and the difference between the actual ground stiffness and the calculated ground stiffness may be large. In the second embodiment, in order to solve this problem, the seismic response analysis is repeated, and the actual ground stiffness is obtained by performing the seismic response analysis from the next time using the shear strain and the cumulative damage obtained in the first time. A method of reducing the error from the ground rigidity by calculation will be described.
[0076]
Hereinafter, as an operation in the second embodiment, a control routine executed by the calculation unit 16 will be described with reference to a flowchart shown in FIG.
[0077]
First, in step 200, the control routine shown in FIG. 2 described above is executed, and the first seismic response analysis is performed. At this time, in step 106 and step 122 shown in FIG. 2, the cumulative damage degree D and the maximum shear strain γ determined for each half wave as shown in FIG._maxAre sequentially stored in the storage unit 14. That is, the cumulative damage degree D and the maximum shear strain γ_maxMemorize the time history and so on. Further, the maximum value of each response value (acceleration, etc.) may be stored.
[0078]
In the next step 202, the cumulative damage degree D and the maximum shear strain γ stored in step 200 are stored._max2 is executed, and the second seismic response analysis is performed. That is, in the second seismic response analysis, in step 106 shown in FIG. 2, the shear stiffness G of the current half wave is changed to the cumulative damage degree D and the maximum shear strain γ in the previous half wave._maxThe maximum shear strain γ closest to the same time or before this time obtained by the first (previous) earthquake response analysis as shown in FIG._maxAnd the cumulative damage degree D at that time. Then, as shown in FIG. 7, the maximum shear strain γ for the next earthquake response analysis is obtained from the obtained shear stiffness G._maxAnd maximum shear stress τ_maxAsk for.
[0079]
Next, in step 204, for example, the maximum shear strain γ obtained in the first seismic response analysis._maxTime shear and maximum shear strain γ obtained by the second seismic response analysis_maxCompare the time history of. If the time histories of the two are substantially the same, it is determined that the first earthquake response analysis is correct, and the next step 206 outputs a response value and the process ends.
[0080]
Note that the determination of whether or not they substantially coincide is, for example, the maximum shear strain γ obtained in the first seismic response analysis at each time._maxAnd the maximum shear strain γ obtained by the second seismic response analysis_maxCan be determined that the time histories of the two have substantially coincided with each other if the difference between the two at each time is all or within a predetermined value within a predetermined value. The comparison in step 204 may be performed using a time history such as the cumulative damage degree D, acceleration, speed, displacement, etc., or the maximum value at all these times.
[0081]
On the other hand, for example, the maximum shear strain γ obtained in the first seismic response analysis_maxTime shear and maximum shear strain γ obtained by the second seismic response analysis_maxIf the time history does not substantially match, the process returns to step 202 to perform the third earthquake response analysis. At this time, the shear stiffness G is the cumulative damage degree D and the maximum shear strain γ at the same time determined by the second seismic response analysis._maxAsk from.
[0082]
Thus, the shear stiffness G is the maximum shear strain γ at the same time determined by the previous earthquake response analysis as shown in FIGS._maxAnd the maximum shear strain γ previously obtained from the cumulative damage degree D_maxTime history and maximum shear strain γ found this time_maxThe earthquake response analysis is repeated until the time history of is almost the same. As a result, even when the earthquake amplitude suddenly increases as in the case of a large earthquake, the ground rigidity error can be reduced, and the seismic response analysis can be performed with high accuracy.
[0083]
【Example】
Next, embodiments of the present invention will be described with reference to the drawings. In this example, an example of the earthquake response analysis method described in the first embodiment will be described.
[0084]
For the analysis cases shown in Table 1 below, earthquake response analysis is performed by the earthquake response analysis apparatus 10 shown in the first embodiment, and these analysis results and analysis results by linear response analysis, time history nonlinear analysis, and dynamic effective stress analysis are analyzed. And compared. As the analysis program, MuDIAN was used for nonlinear analysis including dynamic effective stress analysis, and SHAKE was used for linear analysis and equivalent linear analysis. As a result, the same solution was obtained for the linear analysis, and a fairly close solution was obtained for the equivalent linear analysis.
[0085]
[Table 1]

[0086]
Next, a comparison result between the nonlinear response analysis using the Ramberg-Osgood (RO) model and the effective stress analysis using the Densification model will be described.
[0087]
The ground model used for the study was an actual ground that could be liquefied up to a depth of about 21.4 m, and the model was modeled up to -50 m for analysis. Table 2 shows the ground constant. The standard penetration test is up to −26.3 m, and the physical properties are the same as −26.3 m up to −40 m. The G-γ curve, h-γ curve, and liquefaction strength curve shown in FIG. 3 include liquefaction tests and dynamic deformation tests at −4 m, −6 m, −9 m, −12 m, and −15 m by freezing sampling. The results of were used. Fig. 9 shows G / G₀-A γ curve and a liquefaction strength curve are shown. Table 3 shows parameters of the R-O model and the Densification model. Note that a depth of −21.4 m or more was a linear material, and h = 2%.
[0088]
[Table 2]

[0089]
[Table 3]

[0090]
First, the comparison with the analysis result by the RO model is shown. FIG. 10 shows the surface layer (GL 0.0 mm: Layer 1) and the G.L. L. The response acceleration time history at -14.65 m (Layer 6-7) is shown. In addition, a thick line is an analysis result (analysis case is ER2 in Table 1) by the seismic response analysis method (LEQ) of the present invention, and a thin line is an analysis result by the RO model. FIG. 11 shows a maximum response acceleration distribution map and a maximum shear stress distribution map.
[0091]
As shown in FIG. 10, the waveforms correspond to each other well, but in the RO model, there is a high vibration component and a difference occurs in the peak value. This is also reflected in the maximum response acceleration distribution shown in FIG. L. It is large at -10m to -25m, and small at other points, but it agrees well as a whole.
[0092]
Next, the comparison with the analysis result by Densification model (effective stress analysis) is demonstrated.
[0093]
FIG. 12 shows the liquefaction rate. In order from the top, Layer 2 (GL -2.3 m to -4.4 m), Layer 5 (GL -9.7 m to -12.5 m), Layer 7 (GL --14.65 m to-) The result of 18.6 m) is shown. In addition, a thick line is the analysis result (A-R1 in Table 1) in the earthquake response analysis method (LEQ) of the present invention, and a thin line is the analysis result in the Densification model. FIG. 13 shows a response acceleration time history, and FIG. 14 shows a maximum response acceleration distribution and a maximum response shear stress distribution diagram.
[0094]
As shown in FIG. 12, in the seismic response analysis method according to the present invention, a step shape is used to calculate the liquefaction rate (excess pore water pressure ratio) for each half wave. And have a good response.
[0095]
According to this, it is expected that the liquefaction rate is about 0.7 just below the surface layer and the rigidity is greatly reduced. Reflecting this, the response acceleration time history shown in FIG. 13 becomes longer in response and becomes shorter in about 10 seconds. Further, it is recognized from the liquefaction rate shown in FIG. 12 and the response acceleration time history of FIG. 13 that the intermediate layer (GL-14.65m: Layer 6-7) has liquefied after 15 seconds. be able to. This result is in good agreement with the analysis result by the Densification model.
[0096]
The maximum response acceleration and the maximum response shear stress are slightly different from the case of the analysis result by the nonlinear analysis in the RO model that does not consider liquefaction, but it can be said that the overall response is good.
[0097]
【The invention's effect】
As described above, according to the present invention, it is possible to predict seismic response and permanent deformation regardless of the skill to be analyzed, and the standardization of seismic response prediction technology has progressed, and construction during liquefaction has progressed. It is possible to objectively evaluate the safety of the structure, and even if the earthquake amplitude suddenly increases as in the case of a large earthquake, the ground stiffness error can be reduced and the earthquake can be accurately performed. Response analysis can be performed. In addition, the present invention can be applied not only to the ground but also to a frame structure, and to a general structure if the computer speed increases, and is useful as a stable analysis method for nonlinear dynamic problems.
[Brief description of the drawings]
FIG. 1 is a schematic block diagram of an earthquake response analyzing apparatus.
FIG. 2 is a flowchart of a control routine executed in a calculation unit in the first embodiment.
FIG. 3 is a diagram for explaining calculation of shear rigidity.
FIG. 4 is a flowchart of a control routine executed in a calculation unit in the second embodiment.
FIG. 5 is a diagram for explaining a first calculation result of shear strain and cumulative damage degree in the second embodiment.
FIG. 6 is a diagram for explaining calculation after the second time of shear rigidity in the second embodiment.
FIG. 7 is a diagram for explaining preparation calculation for the next time in the second embodiment.
FIG. 8 is a diagram for explaining calculation of shear rigidity in the second embodiment.
9A is a diagram showing a G-γ curve, and FIG. 9B is a diagram of a liquefaction strength curve.
FIG. 10 is a diagram showing a response acceleration time history.
FIG. 11 is a diagram showing a maximum response acceleration distribution and a maximum shear stress distribution.
FIG. 12 is a diagram showing a time history of the liquefaction rate.
FIG. 13 is a diagram showing a response acceleration time history.
FIG. 14 is a diagram showing a maximum response acceleration distribution and a maximum shear stress distribution.
[Explanation of symbols]
10 Earthquake response analyzer
12 Operation unit
14 Storage unit
16 Calculation unit
18 Display section

Claims (5)

Displacement information of the model to be analyzed that changes from moment to moment is obtained using the equation of motion based on the mass matrix information, stiffness matrix information, attenuation matrix information, and input earthquake information of the model to be analyzed, and from moment to moment based on the displacement information. In the seismic response analysis method to obtain the time characteristics of the response value related to at least strain and the stress response value that change with
Obtaining the response value that changes from moment to moment based on displacement information obtained by time integration of the equation of motion ,
Obtain a representative value of the response value related to the strain and a representative value of the response value related to the stress obtained in a half wave from when the response value related to the strain becomes a predetermined response value to the next predetermined response value .
Rigidity and damping are obtained based on the obtained representative value relating to the strain and the representative value relating to stress ,
An earthquake response analysis method, wherein the stiffness matrix information and the attenuation matrix information are updated every half wave so as to include the obtained stiffness and damping .   The earthquake response analysis method according to claim 1, wherein the rigidity is obtained based on a representative value obtained immediately before. After obtaining the time characteristic of the response value,
Based on displacement information obtained by time integration of the equation of motion, a response value that changes from moment to moment is obtained,
Obtain a representative value of the response value related to the strain obtained in the half wave from the response value related to the strain to a predetermined response value until the next predetermined response value ,
Obtain rigidity and damping based on the representative value for the strain obtained last time corresponding to the time when the response value for the strain was obtained,
Repetitively updating the stiffness matrix information and the attenuation matrix information for each half wave so as to include the obtained stiffness and damping , to obtain a time characteristic,
If it does not match substantially compares the current calculated time characteristics and time characteristics previously determined, until the time characteristic and the current obtained time characteristic previously obtained substantially match was determined by integrating the equations of motion time Obtaining a response value that changes from moment to moment based on displacement information, obtaining a representative value of the response value related to the strain obtained in the half wave ,
Rigidity and damping are obtained based on a representative value relating to the strain obtained last time corresponding to the time when the response value relating to the strain is obtained, and the stiffness matrix information and the attenuation matrix information are included in the half so as to include the obtained stiffness and damping. 3. The earthquake response analysis method according to claim 2, wherein a time characteristic is obtained by repeating updating for each wave, and a process of comparing the time characteristic obtained last time with the time characteristic obtained this time is repeated.   The earthquake response analysis method according to claim 1, wherein the analysis target model is a ground model. Determine the cumulative damage degree from the representative values for the required meta stress,
The earthquake response analysis method according to claim 4, wherein rigidity is obtained based on a representative value related to the stress and a cumulative damage degree.

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